Trigonometric Identities and Formulas

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Pythagorean Identities
  1. cos 2 x + sin 2 x = 1

  2. 1 + tan 2 x = sec 2 x

  3. cot 2 x + 1 = csc 2 x
Ratios of Trigonometric Functions
  1. tan x = sin x / cos x

  2. cot x = 1 / tan x = cos x / sin x

  3. sec x = 1 / cos x

  4. csc x = 1 / sin x
Sum/Difference of Angles Formulas
  1. cos(x + y)= cos x cos y - sin x sin y

  2. cos(x - y)= cos x cos y + sin x sin y

  3. sin(x + y)= sin x cos y + cos x sin y

  4. sin(x - y)= sin x cos y - cos x sin y

  5. tan(x + y)=(tan x + tan y)/(1 - tan x tan y}

  6. tan(x - y)=(tan x - tan y)/(1 + tan x tan y)
Double Angle Formulas/Identities
  1. cos(2 x)= cos 2 x - sin 2 x

  2. sin(2 x)= 2 sin x cos x

  3. tan (2 x)=(2 tan x)/(1 - tan 2 x)
Half Angle Formulas/Identities
  1. sin (x / 2) = + or - √ (1 - cos x)/ 2)

  2. cos (x / 2) = + or - √ (1 + cos x)/ 2)

  3. tan (x / 2) = + or - √ (1 - cos x)/ (1 + cos x)) = sin x / (1 + cos x) = (1 - cos x)/(sin x)
Power Reducing Formulas/Identities
  1. sin 2 x = 1 / 2 - (1 / 2) cos(2 x)

  2. $\cos^2 x = \dfrac{1}{2}+\dfrac{1}{2}\cos(2x)$

  3. $\sin^3 x = \dfrac{3}{4}\sin x -\dfrac{1}{4}\sin(3x)$

  4. $\cos^3 x = \dfrac{3}{4}\cos x +\dfrac{1}{4}\cos(3x)$
Negative Angle Identities
    Odd Functions

  1. $\sin (-x) = -\sin x$

  2. $\tan (-x) = -\tan x$

  3. $\cot (-x) = -\cot x$

  4. $\csc (-x) = -\csc x$

    Even Functions

  5. $\cos (-x) = \cos x$

  6. $\sec (-x) = \sec x$
Sum (to Product) of Trigonometric Functions
  1. $\sin x + \sin y = 2\sin \dfrac{1}{2}(x+y) \cos \dfrac{1}{2}(x-y)$

  2. $\cos x + \cos y = 2\cos \dfrac{1}{2}(x+y) \cos \dfrac{1}{2}(x-y)$
Difference (to Product) of Trigonometric Functions
  1. $\sin x - \sin y = 2\cos \dfrac{1}{2}(x+y) \sin \dfrac{1}{2}(x-y)$

  2. $\cos x - \cos y = 2\sin \dfrac{1}{2}(x+y) \sin \dfrac{1}{2}(x-y)$
Product (to Sum) of Trigonometric Functions
  1. $\sin x \sin y = \dfrac{1}{2}[\cos(x-y)-\cos(x+y)]$

  2. $\sin x \cos y = \dfrac{1}{2}[\sin(x-y)+\sin(x+y)]$

  3. $\cos x \cos y = \dfrac{1}{2}[\cos(x-y)+\cos(x+y)]$