The Collection of Trigonometric Identities

A mathematical identity is an equation that is true for all values of the variable’s domain.

Let \(\theta\) represent a real numeric variable.

Reciprocal Identities

  1. 1.

    \(\sin{\theta}=\dfrac{1}{\csc{\theta}}\)                   \(\csc{\theta}=\dfrac{1}{\sin{\theta}}\)

  2. 2.

    \(\cos{\theta}=\dfrac{1}{\sec{\theta}}\)                   \(\sec{\theta}=\dfrac{1}{\cos{\theta}}\)

  3. 3.

    \(\tan{\theta}=\dfrac{1}{\cot{\theta}}\)                   \(\cot{\theta}=\dfrac{1}{\tan{\theta}}\)

Pythagorean Identities

  1. 1.

    \(\sin^{2}{\theta}+\cos^{2}{\theta}=1\)

  2. 2.

    \(\tan^{2}{\theta}+1=\sec^{2}{\theta}\)

  3. 3.

    \(1+\cot^{2}{\theta}=\csc^{2}{\theta}\)

Opposite Angle Identities

  1. 1.

    \(\sin{\left(-\theta\right)}=-\sin{\left(\theta\right)}\hskip 56.905512pt\csc{\left(-\theta\right)}=-\csc{\left(\theta\right)}\)

  2. 2.

    \(\cos{\left(-\theta\right)}=\cos{\left(\theta\right)}\hskip 56.905512pt\sec{\left(-\theta\right)}=\sec{\left(\theta\right)}\)

  3. 3.

    \(\tan{\left(-\theta\right)}=-\tan{\left(\theta\right)}\hskip 56.905512pt\cot{\left(-\theta\right)}=-\cot{\left(\theta\right)}\)

Cofunction Identities

”Co-” means compliment.

  1. 1.

    \(\sin{\theta}=\cos{\left(\dfrac{\pi}{2}-\theta\right)}\hskip 56.905512pt\cos{\theta}=\sin{\left(\dfrac{\pi}{2}-\theta\right)}\)

  2. 2.

    \(\sec{\theta}=\csc{\left(\dfrac{\pi}{2}-\theta\right)}\hskip 56.905512pt\csc{\theta}=\sec{\left(\dfrac{\pi}{2}-\theta\right)}\)

  3. 3.

    \(\tan{\theta}=\cot{\left(\dfrac{\pi}{2}-\theta\right)}\hskip 56.905512pt\cot{\theta}=\tan{\left(\dfrac{\pi}{2}-\theta\right)}\)