Trigonometric Identities

Sum and Difference Identities

Sum Identities

$\sin(A + B) = \sin A \cos B + \cos A \sin B$

$\cos(A + B) = \cos A \cos B - \sin A \sin B$

$\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$

Difference Identities

$\sin(A - B) = \sin A \cos B - \cos A \sin B$

$\cos(A - B) = \cos A \cos B + \sin A \sin B$

$\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}$

Double Angle Identities

Sine Double Angle

$\sin 2A = 2 \sin A \cos A$

Cosine Double Angle (Three Forms)

$\cos 2A = \cos^2 A - \sin^2 A$

$\cos 2A = 2\cos^2 A - 1$

$\cos 2A = 1 - 2\sin^2 A$

Tangent Double Angle

$\tan 2A = \frac{2\tan A}{1 - \tan^2 A}$

Half Angle Identities

$\sin\left(\frac{A}{2}\right) = \pm\sqrt{\frac{1 - \cos A}{2}}$

$\cos\left(\frac{A}{2}\right) = \pm\sqrt{\frac{1 + \cos A}{2}}$

$\tan\left(\frac{A}{2}\right) = \pm\sqrt{\frac{1 - \cos A}{1 + \cos A}} = \frac{\sin A}{1 + \cos A} = \frac{1 - \cos A}{\sin A}$

Note: Sign ($\pm$) depends on quadrant of $\frac{A}{2}$

Power-Reducing Identities

$\sin^2 A = \frac{1 - \cos 2A}{2}$

$\cos^2 A = \frac{1 + \cos 2A}{2}$

$\tan^2 A = \frac{1 - \cos 2A}{1 + \cos 2A}$

Product-to-Sum Identities

$\sin A \cos B = \frac{1}{2}[\sin(A + B) + \sin(A - B)]$

$\cos A \sin B = \frac{1}{2}[\sin(A + B) - \sin(A - B)]$

$\cos A \cos B = \frac{1}{2}[\cos(A + B) + \cos(A - B)]$

$\sin A \sin B = \frac{1}{2}[\cos(A - B) - \cos(A + B)]$

Sum-to-Product Identities

$\sin A + \sin B = 2 \sin\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right)$

$\sin A - \sin B = 2 \cos\left(\frac{A + B}{2}\right) \sin\left(\frac{A - B}{2}\right)$

$\cos A + \cos B = 2 \cos\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right)$

$\cos A - \cos B = -2 \sin\left(\frac{A + B}{2}\right) \sin\left(\frac{A - B}{2}\right)$

Cofunction Identities

$\sin\left(\frac{\pi}{2} - A\right) = \cos A$

$\cos\left(\frac{\pi}{2} - A\right) = \sin A$

$\tan\left(\frac{\pi}{2} - A\right) = \cot A$

$\cot\left(\frac{\pi}{2} - A\right) = \tan A$

$\sec\left(\frac{\pi}{2} - A\right) = \csc A$

$\csc\left(\frac{\pi}{2} - A\right) = \sec A$

Negative Angle Identities

$\sin(-A) = -\sin A \quad \text{(odd function)}$

$\cos(-A) = \cos A \quad \text{(even function)}$

$\tan(-A) = -\tan A \quad \text{(odd function)}$

$\cot(-A) = -\cot A \quad \text{(odd function)}$

$\sec(-A) = \sec A \quad \text{(even function)}$

$\csc(-A) = -\csc A \quad \text{(odd function)}$

Triple Angle Identities

$\sin 3A = 3 \sin A - 4 \sin^3 A$

$\cos 3A = 4 \cos^3 A - 3 \cos A$

$\tan 3A = \frac{3 \tan A - \tan^3 A}{1 - 3 \tan^2 A}$

Useful Combinations

$\sin A \pm \cos A$

$\sin A + \cos A = \sqrt{2} \sin\left(A + \frac{\pi}{4}\right)$

$\sin A - \cos A = \sqrt{2} \sin\left(A - \frac{\pi}{4}\right)$

$a \sin A + b \cos A$

$a \sin A + b \cos A = R \sin(A + \varphi)$

where $R = \sqrt{a^2 + b^2}$ and $\tan\varphi = \frac{b}{a}$