# Function Identities

## Trigonometric function

1. $\sin(2A) = 2\sin(A)\cos(A) = \frac{2\tan(A)}{1 + \tan^{2}(A)}$
2. $\cos(2A) = \cos^{2}(A) - \sin^{2}(A) = \frac{1 - \tan^{2}(A)}{1 + \tan^{2}(A)}$
3. $\tan(2A) = \frac{2\tan(A)}{1 - \tan^{2}(A)}$
4. $\sin(3A) = 3\sin(A) - 4\sin^{3}(A)$
5. $\cos(3A) = 4\cos^{3}(A) - 3\cos(A)$
6. $\tan(3A) = \frac{3\tan(A) - \tan^{3}(A)}{1 - 3\tan^{2}(A)}$

## Hyperbolic function

1. $\sinh(x) = \frac{e^{x} - e^{-x}}{2}$
2. $\cosh(x) = \frac{e^{x} + e^{-x}}{2}$
3. $\cosh^{2}(x) - \sinh^{2}(x) = 1$
4. $\cosh^{2}(x) + \sinh^{2}(x) = \cosh(2x)$
5. $\text{sech}^{2}(x) + \tanh^{2}(x) = 1$
6. $\sinh(-x) = - \sinh(x)$
7. $\cosh(-x) = \cosh(x)$
8. $\sinh(2x) = 2\sinh(x)\cosh(x)$
9. $\sinh(x + y) = \sinh(x).\cosh(y) + \cosh(x).\sinh(y)$
10. $\cosh(x + y) = \cosh(x).\cosh(y) + \sinh(x).\sinh(y)$

## Exponential function

1. $i = e^{i\frac{\pi}{2}}$
2. $a^{x} = e^{x\ln(a)}$

Published on May 30, 2021

Last revised on Apr 3, 2022