# tan (A - B)

How to solve tan (A - B) problems: formula, proof, examples, and their solutions.

## Formula

tan (A - B) = (tan A - tan B) / (1 + tan A tan B)

## Proof

Divide both of the numerator and the denominator
by cos A cos B.

The dark gray factors are cancelled.

Then
sin A/cos A = tan A,
sin B/cos B = tan B,
(sin A sin B)/(cos A cos B) = tan A tan B.

Quotient identities

## Example 1

tan 15º = tan (45º - 30º)
= (tan 45º - tan 30º) / (1 + tan 45º tan 30º)

Draw a 45-45-90 triangle and a 30-60-90 triangle.

Tangent: TOA.
So tan 45º = 1.
And tan 30º = 1/√3.

So (tan 45º - tan 30º) / (1 + tan 45º tan 30º)
= (1 - 1/√3) / (1 + 1⋅(1/√3)).

Multiply √3 - 1
to both of the numerator and the denominator.

Rationalizing a denominator

Numerator:
(√3 - 1)(√3 - 1) = 3 - 2⋅√3⋅1 + 1

Square of a difference

## Example 2

Think of the blue angle and the green angle like this.

Then θ = (blue angle) - (green angle).

Tangent: TOA.

So tan (blue angle) = (3 + 1)/2
= 2.

And tan (green angle) = 1/2.

So tan θ = tan [(blue angle) - (green angle)]
= [2 - 1/2] / [1 + 2⋅(1/2)].
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