tan (A + B)

tan (A + B)

How to solve tan (A + B) problems: formula, proof, example, and its solution.

Formula

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

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

Proof

tan (A + B): Proof of the Formula

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

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

tan (A - B)

tan (-B) = -tan B

Trigonometric functions of (-θ)

Example

Find the value of the given expression. tan 15 degrees

tan 105º = tan (60º + 45º)
= (tan 60º + tan 45º) / (1 - tan 60º tan 45º)

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

Tangent: TOA.
So tan 60º = √3.
And tan 45º = 1.

So (tan 60º + tan 45º) / (1 - tan 60º tan 45º)
= (√3 + 1) / (1 - √3⋅1).

Change the denominator 1 - √3 into √3 - 1
and write the (-) sign in front of the fraction.
(to make the calculation easy)

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 sum