Trigonometric Identities

Reciprocal Identities

  • csc⁡θ=1sin⁡θ\csc \theta = \frac{1}{\sin \theta}

  • sin⁡θ=1csc⁡θ\sin \theta = \frac{1}{\csc \theta}

  • sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta}

  • cos⁡θ=1sec⁡θ\cos \theta = \frac{1}{\sec \theta}

  • cot⁡θ=1tan⁡θ\cot \theta = \frac{1}{\tan \theta}

  • tan⁡θ=1cot⁡θ\tan \theta = \frac{1}{\cot \theta}

  • tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}

  • cot⁡θ=cos⁡θsin⁡θ\cot \theta = \frac{\cos \theta}{\sin \theta}

Sum and Difference 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)

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

Double Angle Identities

  • sin⁡(2a)=2sin⁡(a)cos⁡(a)\sin(2a) = 2\sin(a)\cos(a)

  • cos⁡(2a)=cos⁡2(a)−sin⁡2(a)\cos(2a) = \cos^2(a) - \sin^2(a)

  • cos⁡(2a)=2cos⁡2(a)−1\cos(2a) = 2\cos^2(a) - 1

  • cos⁡(2a)=1−2sin⁡2(a)\cos(2a) = 1 - 2\sin^2(a)