# Quotient Identity

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

## 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[tan θ]/[sec^{2} θ] = sin θ cos θ

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)/(sec^{2} 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/(sec^{2} θ) = cos^{2} θ.

So [tan θ]⋅[1/(sec^{2} θ)] = [(sin θ)/(cos θ)]⋅[cos^{2} θ].

Cancel the denominator cos θ

and reduce cos^{2} θ to, (cos^{2} θ)/(cos θ), cos θ.

Let's see what you've solved.

You changed the left side, (tan θ)/(sec^{2} θ),

to the right side, sin θ cos θ.

So the given equation is an identity.

So write

∴ (tan θ)/(sec^{2} θ) = sin θ cos θ.

(∴ means 'therefore'.)

This is the solution of this example.