# Trigonometric Identities and Formulas

 
 Pythagorean Identities cos 2 x + sin 2 x = 1 1 + tan 2 x = sec 2 x cot 2 x + 1 = csc 2 x Ratios of Trigonometric Functions tan x = sin x / cos x cot x = 1 / tan x = cos x / sin x sec x = 1 / cos x csc x = 1 / sin x Sum/Difference of Angles Formulas cos(x + y)= cos x cos y - sin x sin y cos(x - y)= cos x cos y + sin x sin y sin(x + y)= sin x cos y + cos x sin y sin(x - y)= sin x cos y - cos x sin y tan(x + y)=(tan x + tan y)/(1 - tan x tan y} tan(x - y)=(tan x - tan y)/(1 + tan x tan y) Double Angle Formulas/Identities cos(2 x)= cos 2 x - sin 2 x sin(2 x)= 2 sin x cos x tan (2 x)=(2 tan x)/(1 - tan 2 x) Half Angle Formulas/Identities sin (x / 2) = + or - √ (1 - cos x)/ 2) cos (x / 2) = + or - √ (1 + cos x)/ 2) tan (x / 2) = + or - √ (1 - cos x)/ (1 + cos x)) = sin x / (1 + cos x) = (1 - cos x)/(sin x) Power Reducing Formulas/Identities sin 2 x = 1 / 2 - (1 / 2) cos(2 x) $\cos^2 x = \dfrac{1}{2}+\dfrac{1}{2}\cos(2x)$ $\sin^3 x = \dfrac{3}{4}\sin x -\dfrac{1}{4}\sin(3x)$ $\cos^3 x = \dfrac{3}{4}\cos x +\dfrac{1}{4}\cos(3x)$ Negative Angle Identities Odd Functions $\sin (-x) = -\sin x$ $\tan (-x) = -\tan x$ $\cot (-x) = -\cot x$ $\csc (-x) = -\csc x$ Even Functions $\cos (-x) = \cos x$ $\sec (-x) = \sec x$ Sum (to Product) of Trigonometric Functions $\sin x + \sin y = 2\sin \dfrac{1}{2}(x+y) \cos \dfrac{1}{2}(x-y)$ $\cos x + \cos y = 2\cos \dfrac{1}{2}(x+y) \cos \dfrac{1}{2}(x-y)$ Difference (to Product) of Trigonometric Functions $\sin x - \sin y = 2\cos \dfrac{1}{2}(x+y) \sin \dfrac{1}{2}(x-y)$ $\cos x - \cos y = 2\sin \dfrac{1}{2}(x+y) \sin \dfrac{1}{2}(x-y)$ Product (to Sum) of Trigonometric Functions $\sin x \sin y = \dfrac{1}{2}[\cos(x-y)-\cos(x+y)]$ $\sin x \cos y = \dfrac{1}{2}[\sin(x-y)+\sin(x+y)]$ $\cos x \cos y = \dfrac{1}{2}[\cos(x-y)+\cos(x+y)]$