# Quotient Identity

How to use the quotient identity to solve the related problems: formula, 1 example, and its solution.

## Formula

### Formula

An identity is an equation
that is always true.

tan θ = (sin θ)/(cos θ)
is the quotient identity.

This equation is always true.
(for all θ)

## Example

### Solution

To show that the given equation is an identity,

start from the left side,
then derive to the right side.

So first write the left side
(tan θ)/(sec2 θ).

Split the numerator and the denominator.

tan θ = (sin θ)/(cos θ)

The reciprocal of secant is cosine.

Secant: in a Right Triangle

So 1/(sec2 θ) = cos2 θ.

So [tan θ]⋅[1/(sec2 θ)] = [(sin θ)/(cos θ)]⋅[cos2 θ].

Cancel the denominator cos θ
and reduce cos2 θ to, (cos2 θ)/(cos θ), cos θ.

Let's see what you've solved.
You changed the left side, (tan θ)/(sec2 θ),
to the right side, sin θ cos θ.

So the given equation is an identity.

So write
∴ (tan θ)/(sec2 θ) = sin θ cos θ.

(∴ means 'therefore'.)

This is the solution of this example.