Question

Which expression is equivalent to sin(alpha + beta) / cos alpha * cos beta?

A) tan(alpha + beta)
B) sin(alpha + beta)
C) cos(alpha + beta)
D) sec(alpha + beta)

Answer 1

The expression sin(alpha + beta) / cos alpha * cos beta is equivalent to tan(alpha + beta).

Step-by-step explanation:

The expression sin(alpha + beta) / cos alpha * cos beta is equivalent to tan(alpha + beta). This can be derived from the trigonometric identity sin(a + b) = sin a * cos b + cos a * sin b.

By substituting alpha for a and beta for b, we get sin(alpha + beta) = sin alpha * cos beta + cos alpha * sin beta.

Dividing both sides by cos alpha * cos beta gives us sin(alpha + beta) / cos alpha * cos beta = (sin alpha * cos beta + cos alpha * sin beta) / cos alpha * cos beta.

Cancelling out the cos alpha and cos beta terms, we are left with tan(alpha + beta).