Quotient Identity

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



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)




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.