# identities

## Summary of Trigonometric Identities

**Basic Identities**

Click here for the derivation of basic identities.

1. $\sin \theta = \dfrac{1}{\csc \theta}; \,\, \csc \theta = \dfrac{1}{\sin \theta}$

2. $\cos \theta = \dfrac{1}{\sec \theta}; \,\, \sec \theta = \dfrac{1}{\cos \theta}$

3. $\tan \theta = \dfrac{\sin \theta}{\cos \theta} = \dfrac{1}{\cot \theta}$

4. $\cot \theta = \dfrac{\cos \theta}{\sin \theta} = \dfrac{1}{\tan \theta}$

## Derivation of Pythagorean Identities

In reference to the right triangle shown and from the functions of a right triangle:

a/c = sin θ

b/c = cos θ

c/b = sec θ

c/a = csc θ

a/b = tan θ

b/a = cot θ

## Derivation of Sum and Difference of Two Angles

The sum and difference of two angles can be derived from the figure shown below.

## Derivation of the Double Angle Formulas

The Double Angle Formulas can be derived from Sum of Two Angles listed below:

$\sin (A + B) = \sin A \, \cos B + \cos A \, \sin B$ → Equation (1)

$\cos (A + B) = \cos A \, \cos B - \sin A \, \sin B$ → Equation (2)

$\tan (A + B) = \dfrac{\tan A + \tan B}{1 - \tan A \, \tan B}$ → Equation (3)