# List of trigonometric identities
OpenStax groups the fundamental identities into four families: the Pythagorean identities, the even-odd identities, the reciprocal identities, and the quotient identities.[1](https://openstax.org/books/algebra-and-trigonometry/pages/9-1-verifying-trigonometric-identities-and-using-trigonometric-identities-to-simplify-trigonometric-expressions) The NIST Digital Library of Mathematical Functions (DLMF) presents the same material for a complex argument \( z \), organized as addition formulas, squares and products, multiples of the argument, and real and imaginary parts and moduli.[2](https://dlmf.nist.gov/4.21)
| Key fact | Detail |
|---|---|
| Core identity | \( \sin^2 z + \cos^2 z = 1 \) and \( \sec^2 z = 1 + \tan^2 z \) hold for complex \( z \).[2](https://dlmf.nist.gov/4.21) |
| Derivation | The unit circle \( x^2 + y^2 = 1 \) with \( x = \cos\theta \), \( y = \sin\theta \) gives the Pythagorean identity; dividing by \( \cos^2\theta \) or \( \sin^2\theta \) gives the secant and cosecant forms.[3](https://math.libretexts.org/Courses/Cosumnes_River_College/Math_373%3A_Trigonometry_for_Calculus/06%3A_Analytic_Trigonometry/6.01%3A_Basic_Trigonometric_Identities_and_Proof_Techniques) |
| Addition formulas | \( \sin(A \pm B) = \sin A \cos B \pm \cos A \sin B \); the cosine form is \( \cos(A+B) = \cos A \cos B - \sin A \sin B \).[4](https://nucinkis-lab.cc.ic.ac.uk/HELM/workbooks/workbook_4/4_3_trig_id.pdf) • [5](https://web.mit.edu/6.02/www/f2006/handouts/Fourier.pdf) |
| Double angle | \( \cos(2\theta) = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta \); \( \sin(2\theta) = 2\sin\theta\cos\theta \).[6](https://math.libretexts.org/Bookshelves/Precalculus/Precalculus_(Stitz-Zeager)/10%3A_Foundations_of_Trigonometry/10.04%3A_Trigonometric_Identities) |
| Completeness | All trigonometric identities are derivable from the addition formulas plus even/odd symmetry (van den Dries, 1986).[7](https://www.ams.org//journals/proc/1986-096-02/S0002-9939-1986-0818470-6/S0002-9939-1986-0818470-6.pdf) |
| Hyperbolic conversion | Osborn's rule: replace each trigonometric function by its hyperbolic counterpart and flip the sign of every term containing a product of 2, 6, 10, ... sines.[8](https://pennance.us/home/downloads/3152/notes/hyperbolic-trig-comparison.pdf) |
| Numerical caution | \( 1 - \cos^2\theta \) and \( \sqrt{1 - x^2} \) near \( x = 1 \) suffer catastrophic cancellation; the cited Sangi library adds guard bits.[9](https://www.allisone.co.jp/lib/sangi/theory/Float-trigonometric_en.html) |
## Pythagorean, reciprocal, and quotient identities
The main Pythagorean identity comes directly from the unit circle. Since the circle has equation \( x^2 + y^2 = 1 \), substituting \( x = \cos\theta \) and \( y = \sin\theta \) gives \( \cos^2\theta + \sin^2\theta = 1 \).[3](https://math.libretexts.org/Courses/Cosumnes_River_College/Math_373%3A_Trigonometry_for_Calculus/06%3A_Analytic_Trigonometry/6.01%3A_Basic_Trigonometric_Identities_and_Proof_Techniques) The same result follows from [Pythagoras](https://www.edgepedia.org/pythagoras)' theorem applied to a right triangle with hypotenuse 1 and legs \( \cos\theta \) and \( \sin\theta \).[10](https://www.sydney.edu.au/content/dam/students/documents/mathematics-learning-centre/trigonometric-identities.pdf)
**Derived forms.** Dividing the identity by \( \cos^2\theta \) yields \( 1 + \tan^2\theta = \sec^2\theta \), and dividing by \( \sin^2\theta \) yields \( 1 + \cot^2\theta = \csc^2\theta \).[3](https://math.libretexts.org/Courses/Cosumnes_River_College/Math_373%3A_Trigonometry_for_Calculus/06%3A_Analytic_Trigonometry/6.01%3A_Basic_Trigonometric_Identities_and_Proof_Techniques) The reciprocal identities (\( \csc\theta = 1/\sin\theta \), \( \sec\theta = 1/\cos\theta \), \( \cot\theta = 1/\tan\theta \)) and the quotient identities (\( \tan\theta = \sin\theta/\cos\theta \), \( \cot\theta = \cos\theta/\sin\theta \)) follow from the definitions of the functions.[1](https://openstax.org/books/algebra-and-trigonometry/pages/9-1-verifying-trigonometric-identities-and-using-trigonometric-identities-to-simplify-trigonometric-expressions) OpenStax groups the fundamental identities into four families, Pythagorean, even-odd, reciprocal, and quotient, and recommends beginning a verification with the more complicated side of the equation.[1](https://openstax.org/books/algebra-and-trigonometry/pages/9-1-verifying-trigonometric-identities-and-using-trigonometric-identities-to-simplify-trigonometric-expressions)
## Shift, periodicity, and symmetry identities
DLMF Table 4.16.2 tabulates the shift relations for all six functions in radians: \( \sin(-\theta) = -\sin\theta \) and \( \cos(-\theta) = \cos\theta \) (the even-odd symmetry), \( \sin(\pi \pm \theta) = \mp\sin\theta \), \( \cos(\tfrac{\pi}{2} \pm \theta) = \mp\sin\theta \), and \( \tan(\pi \pm \theta) = \pm\tan\theta \), with corresponding entries for cosecant, secant, and cotangent.[11](https://dlmf.nist.gov/4.16)
## Angle addition, double-angle, and half-angle identities
The addition formulas are the engine of the whole list: \( \sin(A+B) = \sin A\cos B + \cos A\sin B \) and \( \sin(A-B) = \sin A\cos B - \cos A\sin B \), with the cosine analog \( \cos(A+B) = \cos A\cos B - \sin A\sin B \).[4](https://nucinkis-lab.cc.ic.ac.uk/HELM/workbooks/workbook_4/4_3_trig_id.pdf) • [5](https://web.mit.edu/6.02/www/f2006/handouts/Fourier.pdf) Setting \( A = B \) gives the double-angle identities in three equivalent cosine forms, \( \cos(2\theta) = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta \), plus \( \sin(2\theta) = 2\sin\theta\cos\theta \) and \( \tan(2\theta) = 2\tan\theta/(1 - \tan^2\theta) \).[6](https://math.libretexts.org/Bookshelves/Precalculus/Precalculus_(Stitz-Zeager)/10%3A_Foundations_of_Trigonometry/10.04%3A_Trigonometric_Identities)
**Half-angle signs.** Solving the double-angle formulas for the half angle gives \( \cos(\theta/2) = \pm\sqrt{(1+\cos\theta)/2} \), \( \sin(\theta/2) = \pm\sqrt{(1-\cos\theta)/2} \), and \( \tan(\theta/2) = \pm\sqrt{(1-\cos\theta)/(1+\cos\theta)} \). The choice of sign depends on the quadrant in which the terminal side of \( \theta/2 \) lies, not on the sign of the original function.[6](https://math.libretexts.org/Bookshelves/Precalculus/Precalculus_(Stitz-Zeager)/10%3A_Foundations_of_Trigonometry/10.04%3A_Trigonometric_Identities) • [12](https://openstax.org/books/precalculus-2e/pages/7-3-double-angle-half-angle-formulas) DLMF records the rational half-angle forms \( \tan(z/2) = (1 - \cos z)/\sin z = \sin z/(1 + \cos z) \), which avoid the square root and its sign ambiguity.[2](https://dlmf.nist.gov/4.21)
**Tangent half-angle substitution.** Setting \( t = \tan(A/2) \) converts rational combinations of sines and cosines into rational functions of \( t \), a technique the Imperial College HELM workbook presents as useful for integration when trigonometric functions are present.[4](https://nucinkis-lab.cc.ic.ac.uk/HELM/workbooks/workbook_4/4_3_trig_id.pdf)
## Product-to-sum and sum-to-product identities
Multiplying out the addition formulas gives the product-to-sum family: \( 2\cos p\cos q = \cos(p+q) + \cos(p-q) \), \( 2\sin p\sin q = \cos(p-q) - \cos(p+q) \), and \( 2\sin p\cos q = \sin(p+q) + \sin(p-q) \).[5](https://web.mit.edu/6.02/www/f2006/handouts/Fourier.pdf) LibreTexts verifies these by expanding the right-hand sides with the angle-addition theorems and notes they are particularly useful in calculus.[6](https://math.libretexts.org/Bookshelves/Precalculus/Precalculus_(Stitz-Zeager)/10%3A_Foundations_of_Trigonometry/10.04%3A_Trigonometric_Identities) Running the same formulas in reverse gives the sum-to-product family, for example \( \cos\alpha + \cos\beta = 2\cos\left(\frac{\alpha+\beta}{2}\right)\cos\left(\frac{\alpha-\beta}{2}\right) \) and \( \sin u + \sin v = 2\sin\left(\frac{u+v}{2}\right)\cos\left(\frac{u-v}{2}\right) \).[6](https://math.libretexts.org/Bookshelves/Precalculus/Precalculus_(Stitz-Zeager)/10%3A_Foundations_of_Trigonometry/10.04%3A_Trigonometric_Identities) • [2](https://dlmf.nist.gov/4.21)
## Euler's formula and the unity of the identity system
[Euler's formula](https://www.edgepedia.org/eulers-formula) \( e^{i\theta} = \cos\theta + i\sin\theta \) is proved by taking real and imaginary parts of the infinite series defining \( e^{i\theta} \), using \( i^2 = -1 \), \( i^3 = -i \), and \( i^4 = 1 \).[13](https://www.maths.tcd.ie/~dwilkins/Courses/2BA1/2BA1s10_0809.pdf) Multiplying \( z = \cos\theta + i\sin\theta \) by \( w = \cos\phi + i\sin\phi \) and comparing real and imaginary parts yields the cosine and sine addition formulas in one step.[13](https://www.maths.tcd.ie/~dwilkins/Courses/2BA1/2BA1s10_0809.pdf) Taking powers of \( e^{i\theta} \) produces the multiple-angle formulas for sine and cosine.[14](https://math.columbia.edu/~woit/eulerformula.pdf) Engineering treatments use the same route: MIT's signal-processing notes derive the sum and product identities from \( e^{j\theta} = \cos\theta + j\sin\theta \), and Rose-Hulman's review derives the product identity \( \cos\theta\cos\phi = \tfrac{1}{2}[\cos(\theta+\phi) + \cos(\theta-\phi)] \) by substituting Euler's identities.[5](https://web.mit.edu/6.02/www/f2006/handouts/Fourier.pdf) • [15](https://www.rose-hulman.edu/class/ee/doering/From_Novell/EC300/Black/Documents/trig.pdf)
**Logical structure.** The dependency among the families has been made precise. Van den Dries's 1986 completeness theorem formalizes the identities through the functional equations \( c(x+y) = c(x)c(y) - s(x)s(y) \) and \( s(x+y) = c(x)s(y) + s(x)c(y) \), together with \( c(-x) = c(x) \) and \( s(-x) = -s(x) \), and shows that all trigonometric identities follow from these addition formulas.[7](https://www.ams.org//journals/proc/1986-096-02/S0002-9939-1986-0818470-6/S0002-9939-1986-0818470-6.pdf) In a complementary direction, Zimba's proof shows the angle-difference identities for sine and cosine are independent of the [Pythagorean theorem](https://www.edgepedia.org/pythagorean-theorem) and the Pythagorean identity, and that the double-angle identities can themselves be used to prove the Pythagorean identity, so the usual textbook ordering is one possible arrangement rather than the only one.[16](https://fermatslibrary.com/s/a-new-approach-to-proving-the-pythagorean-theorem)
## By the numbers: numerical stability and computation
Identities that are exact in mathematics can lose digits in floating point. When \( |\theta| \) is small, \( \cos\theta \approx 1 \), so computing \( 1 - \cos^2\theta \) discards most of the effective bits; the Sangi library adds guard bits proportional to the working precision as a countermeasure, and does the same for \( \sqrt{1 - x^2} \) when \( |x| \) is close to 1.[9](https://www.allisone.co.jp/lib/sangi/theory/Float-trigonometric_en.html) The same caution applies at the other end of the range: in binary64 arithmetic the bad cases for correctly rounded sine and cosine are not known for arguments larger than \( 2^{11} \), so implementations either restrict correct rounding to smaller arguments or use Ziv's multi-step technique; and for very large arguments the absolute error of the input may be non-negligible compared with \( \pi \), so the computed output can be far from the mathematically intended value.[17](https://dl.acm.org/doi/10.1145/3747840) A 2025 preprint extends this program, using range reduction that maps a 32-bit float input to a reduced argument to build fast, correctly rounded trigonometric functions.[18](https://ar5iv.labs.arxiv.org/html/2510.13426)
Multiple-angle identities also serve computation directly: one arbitrary-precision scheme maps results back to the original argument range by applying the triple-angle identity \( \sin(3y) = 3\sin(y) - 4\sin^3(y) \) \( k \) times after trisecting the argument.[19](https://www.hvks.com/Numerical/Downloads/HVE%20Fast%20Trigonometric%20functions%20for%20arbitrary%20precision.pdf)
## Inverse, hyperbolic, and multiple-angle extensions
**Inverse functions.** Branch conventions fix the values of the inverse functions and produce identities such as \( \cos^{-1}(-x) = \pi - \cos^{-1}x \), \( \tan^{-1}(-x) = -\tan^{-1}x \), and the cofunction relation \( \sin^{-1}x = \pi/2 - \cos^{-1}x \).[20](https://victorliu.info/pdfs/trig_identities.pdf)
**Hyperbolic analogs.** Euler's formula gives \( \cos t = (e^{it} + e^{-it})/2 \) and \( \sin t = (e^{it} - e^{-it})/(2i) \), and the hyperbolic functions satisfy \( \cosh x = \cos(ix) \) and \( \sinh x = -i\sin(ix) \).[8](https://pennance.us/home/downloads/3152/notes/hyperbolic-trig-comparison.pdf) Osborn's rule converts a circular identity involving integer powers of sine and cosine into a hyperbolic one: change every sine to sinh and cosine to cosh, and switch the sign of every term containing a product of 2, 6, 10, ... sines.[8](https://pennance.us/home/downloads/3152/notes/hyperbolic-trig-comparison.pdf) The rule follows from two corollaries of [Euler's theorem](https://www.edgepedia.org/eulers-theorem), \( \sin(i\theta) = i\sinh(\theta) \) and \( \cos(i\theta) = \cosh(\theta) \).[21](https://www.johndcook.com/blog/2024/08/20/osborn-rule/) For example, \( \tan(2\theta) = 2\tan\theta/(1 - \tan^2\theta) \) becomes \( \tanh(2\theta) = 2\tanh\theta/(1 + \tanh^2\theta) \).[21](https://www.johndcook.com/blog/2024/08/20/osborn-rule/) In multiple-angle conversions the sign bookkeeping needs care: a term with \( \sin^3\theta \) loses its minus sign because two sines are multiplied together, while a term with \( \sin^5\theta \) changes sign twice and keeps its sign.[22](https://www.johndcook.com/blog/2024/12/07/multiple-angles/) MathWorld notes the hyperbolic functions satisfy many identities analogous to the trigonometric ones, inferable through Osborn's rule.[23](https://mathworld.wolfram.com/HyperbolicFunctions.html)
**Complex arguments and triangle laws.** DLMF gives the complex-argument forms \( \sin z = \sin x\cosh y + i\cos x\sinh y \) and \( \cos z = \cos x\cosh y - i\sin x\sinh y \), which link the circular and hyperbolic systems directly.[2](https://dlmf.nist.gov/4.21) In the hyperbolic plane, the law of sines reads \( \sinh a : \sinh b : \sinh c = \sin\alpha : \sin\beta : \sin\gamma \), pairing hyperbolic sines of sides with circular sines of angles, and the law of cosines for angles is \( \cos\gamma = -\cos\alpha\cos\beta + \sin\alpha\sin\beta\cosh c \).[24](https://math.bme.hu/~ghorvath/hyperbolicplane2.pdf) The side form is \( \cosh c = \cosh a\cosh b - \sinh a\sinh b\cos C \) for any triangle in the hyperbolic plane.[25](https://www.ms.uky.edu/~droyster/courses/spring08/math6118/Classnotes/Chapter12.pdf)
## Applications and recent developments
The product-to-sum identities are working tools in [Fourier analysis](https://www.edgepedia.org/fourier-analysis), where MIT's coursework uses them to expand products of sines and cosines into sums of single-frequency terms, and the Fourier synthesis equation expresses a general time function as a weighted combination of exponentials of all frequencies.[5](https://web.mit.edu/6.02/www/f2006/handouts/Fourier.pdf) MIT's 6.3000 signal-processing course lists the same identities, such as \( \sin(a+b) + \sin(a-b) = 2\sin a\cos b \), as standard equipment for trigonometric [Fourier series](https://www.edgepedia.org/fourier-series).[26](https://sigproc.mit.edu/_static/fall26/recitations/Fourier_Series_Trig_Form-handout.pdf)
**Automated reasoning.** The Trigo benchmark, released in October 2023, formalizes trigonometric expression reduction in the Lean proof language from manually annotated high-school exercises, and its experiments show the tasks challenge advanced language models including GPT-4.[27](https://ar5iv.labs.arxiv.org/html/2310.10180) Earlier, the AutoTrig deep-learning prover produced proof steps nearly as short as breadth-first search at about one-thousandth of the time cost, outperforming Sympy, Matlab, and human solvers on a synthetic dataset; its rule set rests on eight basic identity families, including the four product-to-sum formulas and the four addition and subtraction formulas for sine and cosine.[28](https://ar5iv.labs.arxiv.org/html/2207.06679)
**Recent discovery.** New identity families continue to be found. A 2025 constrained PSLQ search produced new 5- and 6-term Machin-like arctangent identities with record-low Lehmer measures of \( \lambda = 1.4572 \) and \( \lambda = 1.3291 \).[29](https://arxiv.org/html/2508.08307) A 2026 preprint expresses a Ramanujan identity in polar coordinates so that its proof reduces to verifying an elementary trigonometric identity, and derives variations on the original.[30](https://arxiv.org/abs/2603.07793v1)
## Common errors and proof technique
Two warnings recur in the teaching literature. First, the \( \pm \) in a half-angle formula is fixed by the quadrant of \( \theta/2 \), not by the sign of the original trigonometric function, and both signs are valid in general.[12](https://openstax.org/books/precalculus-2e/pages/7-3-double-angle-half-angle-formulas) Second, when verifying an identity, the recommended technique is to treat each side separately rather than perform operations between the two sides, such as squaring both sides.[3](https://math.libretexts.org/Courses/Cosumnes_River_College/Math_373%3A_Trigonometry_for_Calculus/06%3A_Analytic_Trigonometry/6.01%3A_Basic_Trigonometric_Identities_and_Proof_Techniques) On the derivation side, textbook treatments differ in emphasis: engineering reviews prove Euler's identities by expanding both sides in power series and comparing, then derive everything else from them,[15](https://www.rose-hulman.edu/class/ee/doering/From_Novell/EC300/Black/Documents/trig.pdf) while precalculus texts derive half-angle formulas from the double-angle formulas[12](https://openstax.org/books/precalculus-2e/pages/7-3-double-angle-half-angle-formulas) and geometry-based texts start from the unit circle.[3](https://math.libretexts.org/Courses/Cosumnes_River_College/Math_373%3A_Trigonometry_for_Calculus/06%3A_Analytic_Trigonometry/6.01%3A_Basic_Trigonometric_Identities_and_Proof_Techniques)
## References
1. [Verifying Trigonometric Identities, OpenStax Algebra and Trigonometry 9.1](https://openstax.org/books/algebra-and-trigonometry/pages/9-1-verifying-trigonometric-identities-and-using-trigonometric-identities-to-simplify-trigonometric-expressions)
2. [DLMF §4.21 Identities, NIST Digital Library of Mathematical Functions](https://dlmf.nist.gov/4.21)
3. [Basic Trigonometric Identities and Proof Techniques, Mathematics LibreTexts 6.1](https://math.libretexts.org/Courses/Cosumnes_River_College/Math_373%3A_Trigonometry_for_Calculus/06%3A_Analytic_Trigonometry/6.01%3A_Basic_Trigonometric_Identities_and_Proof_Techniques)
4. [HELM Workbook 4.3: Trigonometric Identities, Imperial College London](https://nucinkis-lab.cc.ic.ac.uk/HELM/workbooks/workbook_4/4_3_trig_id.pdf)
5. [Fourier Series and Fourier Transforms, MIT 6.02 handout](https://web.mit.edu/6.02/www/f2006/handouts/Fourier.pdf)
6. [Trigonometric Identities, Mathematics LibreTexts 10.4 (Stitz–Zeager)](https://math.libretexts.org/Bookshelves/Precalculus/Precalculus_(Stitz-Zeager)/10%3A_Foundations_of_Trigonometry/10.04%3A_Trigonometric_Identities)
7. [L. van den Dries (1986). A completeness theorem for trigonometric identities. Proceedings of the AMS 96(2)](https://www.ams.org//journals/proc/1986-096-02/S0002-9939-1986-0818470-6/S0002-9939-1986-0818470-6.pdf)
8. [Trigonometric and Hyperbolic Functions – A Comparison, course notes](https://pennance.us/home/downloads/3152/notes/hyperbolic-trig-comparison.pdf)
9. [Float Trigonometric Function Algorithms, Sangi Algorithm Guide](https://www.allisone.co.jp/lib/sangi/theory/Float-trigonometric_en.html)
10. [Trigonometric Identities, University of Sydney Mathematics Learning Centre](https://www.sydney.edu.au/content/dam/students/documents/mathematics-learning-centre/trigonometric-identities.pdf)
11. [DLMF §4.16 Elementary Properties, NIST](https://dlmf.nist.gov/4.16)
12. [Double-Angle, Half-Angle, and Reduction Formulas, OpenStax Precalculus 2e 7.3](https://openstax.org/books/precalculus-2e/pages/7-3-double-angle-half-angle-formulas)
13. [Section 10: Trigonometric Identities, Complex, Trinity College Dublin 2BA1 notes](https://www.maths.tcd.ie/~dwilkins/Courses/2BA1/2BA1s10_0809.pdf)
14. [Euler's Formula, Columbia University mathematics notes](https://math.columbia.edu/~woit/eulerformula.pdf)
15. [Trigonometry Review for Signals and Systems, Rose-Hulman ECE](https://www.rose-hulman.edu/class/ee/doering/From_Novell/EC300/Black/Documents/trig.pdf)
16. [A New Approach to Proving the Pythagorean Theorem, Fermat's Library annotation (Zimba)](https://fermatslibrary.com/s/a-new-approach-to-proving-the-pythagorean-theorem)
17. [Correctly Rounded Evaluation of a Function: Why, How, and at What Cost? ACM Computing Surveys](https://dl.acm.org/doi/10.1145/3747840)
18. [Fast Trigonometric Functions using the RLIBM Approach, arXiv 2510.13426](https://ar5iv.labs.arxiv.org/html/2510.13426)
19. [Fast Trigonometric functions for arbitrary precision, HVE white paper](https://www.hvks.com/Numerical/Downloads/HVE%20Fast%20Trigonometric%20functions%20for%20arbitrary%20precision.pdf)
20. [Trigonometric identities reference sheet, Victor Liu](https://victorliu.info/pdfs/trig_identities.pdf)
21. [Osborn's rule for turning trig identities into hyperbolic trig identities, John D. Cook (2024)](https://www.johndcook.com/blog/2024/08/20/osborn-rule/)
22. [Multiple angles and Osborn's rule, John D. Cook (2024)](https://www.johndcook.com/blog/2024/12/07/multiple-angles/)
23. [Hyperbolic Functions, Wolfram MathWorld](https://mathworld.wolfram.com/HyperbolicFunctions.html)
24. [The hyperbolic plane, BME course notes](https://math.bme.hu/~ghorvath/hyperbolicplane2.pdf)
25. [Hyperbolic geometry class notes, Chapter 12, University of Kentucky](https://www.ms.uky.edu/~droyster/courses/spring08/math6118/Classnotes/Chapter12.pdf)
26. [Fourier Series Trig Form handout, MIT 6.3000 Signal Processing](https://sigproc.mit.edu/_static/fall26/recitations/Fourier_Series_Trig_Form-handout.pdf)
27. [Trigo: Benchmarking Formal Mathematical Proof Reduction for Generative Language Models, arXiv 2310.10180](https://ar5iv.labs.arxiv.org/html/2310.10180)
28. [Learning to Prove Trigonometric Identities (AutoTrig), arXiv 2207.06679](https://ar5iv.labs.arxiv.org/html/2207.06679)
29. [Constrained PSLQ Search for Machin-like Identities Achieving Record-Low Lehmer Measures, arXiv 2508.08307](https://arxiv.org/html/2508.08307)
30. [A trigonometric approach to an identity by Ramanujan, arXiv 2603.07793](https://arxiv.org/abs/2603.07793v1)
---
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