Quotient Identities

Quotient Identities

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

Identities

tan = sin/cos, cot = cos/sin = 1/tan

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

Quotient Identities: Proof of the Identities

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

csc = 1/sin, sec = 1/cos, tan = sin/cos, cot = cos/sin

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

Show that the given equation is an identity. (tan theta)/(sec^2 theta) = (sin theta)(cos theta)

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/sec2 θ into cos2 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 / sec2 θ = sin θ cos θ is an identity.

This is the way of proving an identity.