Trigonometric Identities and Formulas

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. \(\tan x = \dfrac{\sin x}{\cos x}\)
  2. \(\cot x = \dfrac{1}{\tan x} = \dfrac{\cos x}{\sin x}\)
  3. \(\sec x = \dfrac{1}{\cos x}\)
  4. \(\csc x = \dfrac{1}{\sin x}\)

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. \(\tan(x + y) = \dfrac{\tan x + \tan y}{1 - \tan x \tan y}\)
  6. \(\tan(x - y) = \dfrac{\tan x - \tan y}{1 + \tan x \tan y}\)

Double Angle Formulas/Identities

  1. \(\cos(2x) = \cos^2 x - \sin^2 x\)
  2. \(\sin(2x) = 2 \sin x \cos x\)
  3. \(\tan(2x) = \dfrac{2 \tan x}{1 - \tan^2 x}\)

Half Angle Formulas/Identities

  1. \(\sin\left(\dfrac{x}{2}\right) = \pm \sqrt{\dfrac{1 - \cos x}{2}}\)
  2. \(\cos\left(\dfrac{x}{2}\right) = \pm \sqrt{\dfrac{1 + \cos x}{2}}\)
  3. \(\tan\left(\dfrac{x}{2}\right) = \pm \sqrt{\dfrac{1 - \cos x}{1 + \cos x}} = \dfrac{\sin x}{1 + \cos x} = \dfrac{1 - \cos x}{\sin x}\)

Power Reducing Formulas/Identities

  1. \(\sin^2 x = \dfrac{1}{2} - \dfrac{1}{2}\cos(2x)\)
  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
  1. \(\cos(-x) = \cos x\)
  2. \(\sec(-x) = \sec x\)

Sum (to Product) of Trigonometric Functions

  1. \(\sin x + \sin y = 2\sin\left(\dfrac{x+y}{2}\right) \cos\left(\dfrac{x-y}{2}\right)\)
  2. \(\cos x + \cos y = 2\cos\left(\dfrac{x+y}{2}\right) \cos\left(\dfrac{x-y}{2}\right)\)

Difference (to Product) of Trigonometric Functions

  1. \(\sin x - \sin y = 2\cos\left(\dfrac{x+y}{2}\right) \sin\left(\dfrac{x-y}{2}\right)\)
  2. \(\cos x - \cos y = -2\sin\left(\dfrac{x+y}{2}\right) \sin\left(\dfrac{x-y}{2}\right)\)

Product (to Sum) of Trigonometric Functions

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