Trigonometric Identities
Shared 1/22/2026•25 views
### Reciprocal and Quotient Identities - $\sin x = \frac{1}{\csc x}$ - $\cos x = \frac{1}{\sec x}$ - $\tan x = \frac{1}{\cot x} = \frac{\sin x}{\cos x}$ - $\csc x = \frac{1}{\sin x}$ - $\sec x = \frac{1}{\cos x}$ - $\cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x}$ ### Pythagorean Identities - $\sin^2 x + \cos^2 x = 1$ - $1 + \tan^2 x = \sec^2 x$ - $1 + \cot^2 x = \csc^2 x$ ### Angle Addition and Subtraction Identities - $\sin(A + B) = \sin A \cos B + \cos A \sin B$ - $\sin(A - B) = \sin A \cos B - \cos A \sin B$ - $\cos(A + B) = \cos A \cos B - \sin A \sin B$ - $\cos(A - B) = \cos A \cos B + \sin A \sin B$ - $\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$ - $\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}$ ### Sum-to-Product Identities - $\sin P + \sin Q = 2 \sin\left(\frac{P+Q}{2}\right) \cos\left(\frac{P-Q}{2}\right)$ - $\sin P - \sin Q = 2 \cos\left(\frac{P+Q}{2}\right) \sin\left(\frac{P-Q}{2}\right)$ - $\cos P + \cos Q = 2 \cos\left(\frac{P+Q}{2}\right) \cos\left(\frac{P-Q}{2}\right)$ - $\cos P - \cos Q = -2 \sin\left(\frac{P+Q}{2}\right) \sin\left(\frac{P-Q}{2}\right)$ ### Double-Angle Identities - $\sin(2x) = 2 \sin x \cos x$ - $\cos(2x) = \cos^2 x - \sin^2 x$ - $\cos(2x) = 2 \cos^2 x - 1$ - $\cos(2x) = 1 - 2 \sin^2 x$ - $\tan(2x) = \frac{2 \tan x}{1 - \tan^2 x}$ ### Limit-Critical Trigonometric Results - $\lim_{x \to 0} \frac{\sin x}{x} = 1$ - $\lim_{x \to 0} \frac{\tan x}{x} = 1$ - $\lim_{x \to 0} \frac{1 - \cos x}{x} = 0$ ### Even and Odd Trigonometric Identities - $\sin(-x) = -\sin x$ (Odd) - $\cos(-x) = \cos x$ (Even) - $\tan(-x) = -\tan x$ (Odd) - $\csc(-x) = -\csc x$ (Odd) - $\sec(-x) = \sec x$ (Even) - $\cot(-x) = -\cot x$ (Odd)