Trigonometric identities

Double angle formulas

• $\cos(2x) = \cos^2 x- \sin^2 x$.

• $\cos(2x) = 2\cos^2 x - 1$.

• $\cos(2x) = 1 - 2\sin^2x$.

• $\sinh(2x) = 2 \sinh x \cosh x$.

• $\cosh(2x) = 2 \cosh^2 x - 1$.

Sine and cosine addition formulas

• $A \cos (\theta + \alpha) + B \sin (\theta + \beta) = \sqrt{ \left( A \cos \alpha + B \sin \beta \right)^2 + \left( A \sin \alpha - B \cos \beta \right)^2 } \cdot \cos \left( \theta + \tan^{-1} \left[ \frac{A \sin \alpha - B \cos \beta}{A \cos \alpha + B \sin \beta} \right] \right)$

• $A \cos \theta + B \sin \theta = \sqrt{A^2 +B^2} \cdot \cos \left( \theta - \tan^{-1} \frac BA \right)$.

Proof.

Angle addition formulas

• $\cos(a+b) = \cos a \cos b - \sin a \sin b$.

• $\sin(a+b) = \sin a \cos b + \cos a \sin b$.