Exercise 30 Page 963

Before we can verify the given identity, we need to first consider which Trigonometric Identities will be useful in this equation. Let's recall the Angle Difference Identity for the sine. sin ( A - B ) = sin A cos B - cos A sin BWith this relationship in mind, let's verify the identity!

- sin ( θ - π/2 ) ? = cos θ

sin(α-β)=sin(α)cos(β)-cos(α)sin(β)

- ( sin θ cos π/2 - cos θ sin π/2 ) ? = cos θ

Distr

Distribute -1

- sin θ cos π/2 + cos θ sin π/2 ? = cos θ

cos π/2 = 0

- sin θ (0) + cos θ sin π/2 ? = cos θ

ZeroPropMult

Zero Property of Multiplication

0 + cos θ sin π/2 ? = cos θ

IdPropAdd

Identity Property of Addition

cos θ sin π/2? = cos θ

sin π/2 = 1

cos θ (1) ? = cos θ

IdPropMult

Identity Property of Multiplication

cos θ = cos θ

We started with the left-hand side of the given identity, modified it using other known relationships, and arrived at the right-hand side.

- sin ( θ - π/2 ) ? = cos θ

Simplify LHS

cos θ = cos θ ✓

We have verified the identity!