Trigonometric Identities

Trigonometric Identities

Reciprocal Identities

  • 1sin⁡x=csc⁡x\frac{1}{\sin{x}} = \csc{x}, which also implies 1csc⁡x=sin⁡x\frac{1}{\csc{x}} = \sin{x}

  • csc⁡x=1sin⁡x\csc{x} = \frac{1}{\sin{x}}

  • sin⁡x=1csc⁡x\sin{x} = \frac{1}{\csc{x}}

  • sec⁡x=1cos⁡x\sec{x} = \frac{1}{\cos{x}}

  • cos⁡x=1sec⁡x\cos{x} = \frac{1}{\sec{x}}

  • cot⁡x=1tan⁡x\cot{x} = \frac{1}{\tan{x}}

  • tan⁡x=1cot⁡x\tan{x} = \frac{1}{\cot{x}}

Quotient Identities

  • tan⁡x=sin⁡xcos⁡x\tan{x} = \frac{\sin{x}}{\cos{x}}

  • cot⁡x=cos⁡xsin⁡x\cot{x} = \frac{\cos{x}}{\sin{x}}

Pythagorean Identities

  • sin⁡2θ+cos⁡2θ=1\sin^2{\theta} + \cos^2{\theta} = 1

  • 1+tan⁡2θ=sec⁡2θ1 + \tan^2{\theta} = \sec^2{\theta}

  • 1+cot⁡2θ=csc⁡2θ1 + \cot^2{\theta} = \csc^2{\theta}

Other Identities

  • 1−cos⁡2x=sin⁡2x1 - \cos^2{x} = \sin^2{x}

  • 1+tan⁡2x=sec⁡2x1 + \tan^2{x} = \sec^2{x}

  • 1cos⁡2x=sec⁡2x\frac{1}{\cos^2{x}} = \sec^2{x}

  • cos⁡xsin⁡x=cot⁡x\frac{\cos{x}}{\sin{x}} = \cot{x}

  • 1sin⁡x=csc⁡x\frac{1}{\sin{x}} = \csc{x}

  • 1cos⁡2x=sec⁡2x\frac{1}{\cos^2 x} = \sec^2 x