Trigonometric Identities


Reciprocal Identities

  • \(\displaystyle \sin u = \frac{1}{\csc u}\)
  • \(\displaystyle \cos u = \frac{1}{\sec u}\)
  • \(\displaystyle \tan u = \frac{1}{\cot u}\)
  • \(\displaystyle \csc u = \frac{1}{\sin u}\)
  • \(\displaystyle \sec u = \frac{1}{\cos u}\)
  • \(\displaystyle \cot u = \frac{1}{\tan u}\)
Quotient Identities

  • \(\displaystyle \tan u = \frac{\sin u}{\cos u}\)
  • \(\displaystyle \cot u = \frac{\cos u}{\sin u}\)
Pythagorean Identities

  • \(\displaystyle \sin^2 u + \cos^2 u = 1\)
  • \(\displaystyle 1 + \tan^2 u = \sec^2u\)
  • \(\displaystyle 1 + \cot^2 u = \csc^2u\)
Cofunction Identities

  • \(\displaystyle \sin\left( \frac{\pi}{2} – u\right) = \cos u\)
  • \(\displaystyle \cot\left( \frac{\pi}{2} – u\right) = \tan u\)
  • \(\displaystyle \cos\left( \frac{\pi}{2} – u\right) = \sin u\)
  • \(\displaystyle \sec\left( \frac{\pi}{2} – u\right) = \csc u\)
  • \(\displaystyle \tan\left( \frac{\pi}{2} – u\right) = \cot u\)
  • \(\displaystyle \csc\left( \frac{\pi}{2} – u\right) = \sec u\)
Even/Odd Identities

  • \(\displaystyle \sin(-u) = -\sin u\)
  • \(\displaystyle \cot(-u) = -\cot u\)
  • \(\displaystyle \cos(-u) = \cos u\)
  • \(\displaystyle \sec(-u) = \sec u\)
  • \(\displaystyle \tan(-u) = -\tan u\)
  • \(\displaystyle \cot(-u) = -\cot u\)
Sum and Difference Formulas

  • \(\displaystyle \sin(u \pm v) = \sin u \cos v \pm \cos u \sin v\)
  • \(\displaystyle \cos(u \pm v) = \cos u \cos v \mp \sin u \sin v\)
  • \( \tan(u \pm v) = \dfrac{\tan u \pm \tan v}{1 \mp \tan u \tan v}\)
Double-Angle Formulas

  • \(\displaystyle \sin 2u = 2 \sin u \cos u\)
  • \(\displaystyle \begin{aligned} \cos 2u & = \cos^2 u – \sin^2 u \\
    & = 2 \cos^2 u -1 \\
    & = 1-2\sin^2 u
    \end{aligned}
    \)
  • \(\displaystyle \tan 2u = \frac{2 \tan u}{1-\tan^2 u}\)
Power-Reducing Formulas

  • \(\displaystyle \sin^2u = \frac{1-\cos 2u}{2}\)
  • \(\displaystyle \cos^2u = \frac{1+\cos 2u}{2}\)
  • \(\displaystyle \tan^2 u = \frac{1-\cos 2u}{1+ \cos 2u}\)
Sum-to-Product Formulas

  • \(\displaystyle \sin u + \sin v = 2 \sin\left(\frac{u+v}{2}\right)\cos\left(\frac{u-v}{2}\right)\)
  • \(\displaystyle \sin u – \sin v = 2 \cos\left(\frac{u+v}{2}\right)\sin\left(\frac{u-v}{2}\right)\)
  • \(\displaystyle \cos u + \cos v = 2 \cos\left(\frac{u+v}{2}\right)\cos\left(\frac{u-v}{2}\right)\)
  • \(\displaystyle \cos u – \cos v = -2 \sin\left(\frac{u+v}{2}\right)\sin\left(\frac{u-v}{2}\right)\)
Product-to-Sum Formulas

  • \(\displaystyle \sin u \sin v = \frac{1}{2}[\cos(u-v) – \cos(u+v)]\)
  • \(\displaystyle \cos u \cos v = \frac{1}{2}[\cos(u-v) + \cos(u+v)]\)
  • \(\displaystyle \sin u \cos v = \frac{1}{2}[\sin(u+v) + \sin(u-v)]\)
  • \(\displaystyle \cos u \sin v = \frac{1}{2}[\sin(u+v) – \sin(u-v)]\)