Trigonometric Identities

Basic Definitions

  • tan x=sin x+cos x
  • csc u = \frac{1}{sin u}
  • sec u = \frac{1}{cos u}
  • cot u = \frac{1}{tan u}

Pythagorean Identities

  • sin^2u+cos^2u=1
  • 1+tan^2u=sec^2u
  • 1+cot^2u=csc^2u

Co-Function Identities

  • sin(u\pm\frac{\pi}{2}) = cos u
  • csc(u\pm\frac{\pi}{2}) = sec u
  • tan(u\pm\frac{\pi}{2}) = cot u

Sum-Difference Formulas

  • sin(u\pm v)=sinucosv\pm cos u sin v
  • cos(u\pm v)=cosucosv\mp sin u sin v
  • tan(u\pm v)=\frac{tan u\pm tan v}{1\mp tan u tan v}

Double Angle Formulas

  • sin(2u)=2sin u cos u
  • cos(2u)=cos^2u-sin^2u
  • tan(2u)=\frac{2tan u}{1-tan^2u}

Power-Reducing/Half Angle Formulas

  • sin^2u=\frac{1-cos(2u)}{2}
  • cos^2u=\frac{1+cos(2u)}{2}
  • tan^2u=\frac{1-cos(2u)}{1+cos(2u)}

Sum-to-Product Formulas

  • sin(u) \pm sin(v) = 2sin(\frac{u\pm v}{2})cos(\frac{u\mp v}{2})
  • cos(u) + cos(v) = 2cos(\frac{u+v}{2})cos(\frac{u-v}{2})
  • cos(u) - cos(v) = -2sin(\frac{u+v}{2})sin(\frac{u-v}{2})

Product-to-Sum Formulas

  • sinusinv = \frac{1}{2}[cos(u-v)-cos(u+v)]
  • cosucosv = \frac{1}{2}[cos(u-v)+cos(u+v)]
  • sinucosv = \frac{1}{2}[sin(u+v)+sin(u-v)]

Hyperbolic

  • cosh^2u-sinh^2u=1
  • 1-tanh^2u=sech^2u
  • sinh2u=cosh^2u+sinh^2u