# Quotient Identities

How to solve the quotient identities problems: identities, proof, example, and its solution.

## Identities

tan *θ* = sin *θ* / cos *θ*

Cotangent is the reciprocal of tangent.

(cot *θ* = 1 / tan *θ*)

So cot *θ* = cos *θ* / sin *θ*.

## Proof

Draw a right triangle like this.

Write the central angle as *θ*.

Sine: SOH.

So sin *θ* = *y*/*r*.

Cosine: CAH.

So cos *θ* = *x*/*r*.

Tangent: TOA.

So tan *θ* = *y*/*x*.

Divide *r* to both of the numerator and the denominator:*y*/*r* / *x*/*r*.

Put sin *θ* = *y*/*r* and cos *θ* = *x*/*r*

into the expression.

Then *y*/*r* / *x*/*r* = sin *θ* / cos *θ*.

## How to Change Trigonometric Functions

When solving trigonometric identity problems,

try to change the trigonometric functions

into sine or cosine functions.

Then most of the problems will be easily solved.

## Example

When solving identity problems,

try to change the left side

to make the right side.

(or vice versa)

Don't try to 'solve' the equation

by doing the operations on both sides.

(like solving equations)

Start from the left side.

Change tan *θ* into sin *θ* / cos *θ*.

And secant is the reciprocal of cosine.

So change 1/sec^{2} *θ* into cos^{2} *θ*.

After cancelling cos *θ*

on both of the numerator and the denominator,

you get sin *θ* cos *θ*,

which is the right side of the given equation.

So tan *θ* / sec^{2} *θ* = sin *θ* cos *θ* is an identity.

This is the way of proving an identity.