# Basic Trigonometric Identities

Trigonometric identities are equations involving the trigonometric functions that are true for every value of the variables involved.

Some of the most commonly used trigonometric identities are derived from the Pythagorean Theorem , like the following:

$\begin{array}{l}{\mathrm{sin}}^{2}\left(x\right)+{\mathrm{cos}}^{2}\left(x\right)=1\\ 1+{\mathrm{tan}}^{2}\left(x\right)={\mathrm{sec}}^{2}\left(x\right)\\ 1+{\mathrm{cot}}^{2}\left(x\right)={\mathrm{csc}}^{2}\left(x\right)\end{array}$

Example 1:

Simplify the expression using trigonometric identities.

$1-\frac{{\mathrm{sin}}^{2}\left(\theta \right)}{{\mathrm{tan}}^{2}\left(\theta \right)}$

Rewrite $\mathrm{tan}$ as $\mathrm{sin}/\mathrm{cos}$ .

$\begin{array}{l}=1-\frac{{\mathrm{sin}}^{2}\left(\theta \right)}{\left(\frac{{\mathrm{sin}}^{2}\left(\theta \right)}{{\mathrm{cos}}^{2}\left(\theta \right)}\right)}\\ =1-{\mathrm{sin}}^{2}\left(\theta \right)\cdot \frac{{\mathrm{cos}}^{2}\left(\theta \right)}{{\mathrm{sin}}^{2}\left(\theta \right)}\\ =1-{\mathrm{cos}}^{2}\left(\theta \right)\end{array}$

Use the fundamental Pythagorean identity, we get

$={\mathrm{sin}}^{2}\left(\theta \right)$

### The Reciprocal identities

$\begin{array}{ccc}\mathrm{sin}\left(x\right)=\frac{1}{\mathrm{csc}\left(x\right)}& \mathrm{cos}\left(x\right)=\frac{1}{\mathrm{sec}\left(x\right)}& \mathrm{tan}\left(x\right)=\frac{1}{\mathrm{cot}\left(x\right)}\\ \mathrm{csc}\left(x\right)=\frac{1}{\mathrm{sin}\left(x\right)}& \mathrm{sec}\left(x\right)=\frac{1}{\mathrm{cos}\left(x\right)}& \mathrm{cot}\left(x\right)=\frac{1}{\mathrm{tan}\left(x\right)}\end{array}$

Example 2:

Show that ${\mathrm{sec}}^{2}\left(\theta \right)+{\mathrm{csc}}^{2}\left(\theta \right)={\mathrm{sec}}^{2}\left(\theta \right)\cdot {\mathrm{csc}}^{2}\left(\theta \right)$ .

### The Quotient Identities

$\begin{array}{l}\mathrm{tan}\left(u\right)=\frac{\mathrm{sin}\left(u\right)}{\mathrm{cos}\left(u\right)}\\ \mathrm{cot}\left(u\right)=\frac{\mathrm{cos}\left(u\right)}{\mathrm{sin}\left(u\right)}\end{array}$

Example 3:

Verify the identity, $\mathrm{cos}\left(\theta \right)+\mathrm{sin}\left(\theta \right)\mathrm{tan}\left(\theta \right)=\mathrm{sec}\left(\theta \right)$

Consider the expression on the left side of the equation.