A trigonometric identity is a mathematical equation that is always true for all values of the variables involved, as long as the expressions are defined. These identities relate the trigonometric functions—sine, cosine, tangent, cotangent, secant, and cosecant—to one another.
Here’s a comprehensive list of trigonometric formulas. These formulas are essential for solving problems in trigonometry, calculus, and physics.
Basic Identities
\( \displaystyle \csc(x) = \frac{1}{\sin(x)}, \quad \sec(x) = \frac{1}{\cos(x)}, \quad \cot(x) = \frac{1}{\tan(x)} \\ \displaystyle \tan(x) = \frac{\sin(x)}{\cos(x)}, \quad \cot(x) = \frac{\cos(x)}{\sin(x)} \)Pythagorean Identities
\( \displaystyle \sin^2(x) + \cos^2(x) = 1 \\ \displaystyle 1 + \tan^2(x) = \sec^2(x) \\ \displaystyle 1 + \cot^2(x) = \csc^2(x) \)Angle Sum and Difference Identities
\( \displaystyle \sin(a \pm b) = \sin(a)\cos(b) \pm \cos(a)\sin(b) \\ \displaystyle \cos(a \pm b) = \cos(a)\cos(b) \mp \sin(a)\sin(b) \\ \displaystyle \tan(a \pm b) = \frac{\tan(a) \pm \tan(b)}{1 \mp \tan(a)\tan(b)} \)Double-Angle Identities
\( \displaystyle \sin(2x) = 2\sin(x)\cos(x) \\ \displaystyle \cos(2x) = \cos^2(x) – \sin^2(x) = 2\cos^2(x) – 1 = 1 – 2\sin^2(x) \\ \displaystyle \tan(2x) = \frac{2\tan(x)}{1 – \tan^2(x)} \)Half-Angle Identities
\( \displaystyle \sin\left(\frac{x}{2}\right) = \pm \sqrt{\frac{1 – \cos(x)}{2}} \\ \displaystyle \cos\left(\frac{x}{2}\right) = \pm \sqrt{\frac{1 + \cos(x)}{2}} \\ \displaystyle \tan\left(\frac{x}{2}\right) = \pm \sqrt{\frac{1 – \cos(x)}{1 + \cos(x)}} = \frac{1 – \cos(x)}{\sin(x)} = \frac{\sin(x)}{1 + \cos(x)} \)Power-Reducing Identities
\( \displaystyle \sin^2(x) = \frac{1 – \cos(2x)}{2} \\ \displaystyle \cos^2(x) = \frac{1 + \cos(2x)}{2} \\ \displaystyle \tan^2(x) = \frac{1 – \cos(2x)}{1 + \cos(2x)} \)Sum-to-Product Identities
\( \displaystyle \sin(a) + \sin(b) = 2\sin\left(\frac{a + b}{2}\right)\cos\left(\frac{a – b}{2}\right) \\ \displaystyle \sin(a) – \sin(b) = 2\cos\left(\frac{a + b}{2}\right)\sin\left(\frac{a – b}{2}\right) \\ \displaystyle \cos(a) + \cos(b) = 2\cos\left(\frac{a + b}{2}\right)\cos\left(\frac{a – b}{2}\right) \\ \displaystyle \cos(a) – \cos(b) = -2\sin\left(\frac{a + b}{2}\right)\sin\left(\frac{a – b}{2}\right) \)Product-to-Sum Identities
\( \displaystyle \sin(a)\sin(b) = \frac{1}{2} [\cos(a – b) – \cos(a + b)] \\ \displaystyle \cos(a)\cos(b) = \frac{1}{2} [\cos(a – b) + \cos(a + b)] \\ \displaystyle \sin(a)\cos(b) = \frac{1}{2} [\sin(a + b) + \sin(a – b)] \\ \)Even-Odd Identities
\( \displaystyle \sin(-x) = -\sin(x) \\ \displaystyle \cos(-x) = \cos(x) \\ \displaystyle \tan(-x) = -\tan(x) \)Cofunction Identities
\( \displaystyle \sin\left(\frac{\pi}{2} – x\right) = \cos(x), \quad \cos\left(\frac{\pi}{2} – x\right) = \sin(x) \\ \displaystyle \tan\left(\frac{\pi}{2} – x\right) = \cot(x), \quad \cot\left(\frac{\pi}{2} – x\right) = \tan(x) \\ \displaystyle \sec\left(\frac{\pi}{2} – x\right) = \csc(x), \quad \csc\left(\frac{\pi}{2} – x\right) = \sec(x) \)Inverse Trigonometric Identities
\( \displaystyle \sin^{-1}(-x) = -\sin^{-1}(x) \\ \displaystyle \cos^{-1}(-x) = \pi – \cos^{-1}(x) \\ \displaystyle \tan^{-1}(-x) = -\tan^{-1}(x) \)Miscellaneous Identities
\( \displaystyle \sin(3x) = 3\sin(x) – 4\sin^3(x) \\ \displaystyle \cos(3x) = 4\cos^3(x) – 3\cos(x) \\ \displaystyle \tan(3x) = \frac{3\tan(x) – \tan^3(x)}{1 – 3\tan^2(x)} \)Please let me know in the comments if you find any error in any formula.