Exercise 31 Page 963

Consider how the given equation can be modified using all of the various Trigonometric Identities.

See solution.

Practice makes perfect

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 Cosecant Identity, the Angle Difference Identity for sine, and the Secant Identity.

Cosecant Identity	csc A=1/sin A
Angle Difference Identity for Sine	sin (A-B)= sin A cos B-cos A sin B
Secant Identity	sec A= 1/cos A

With these relationships in mind, let's verify the identity!

csc ( θ - π/2 ) ? = sec θ

csc(θ) = 1/sin(θ)

1/sin ( θ - π2 ) ? = sec θ

sin(α-β)=sin(α)cos(β)-cos(α)sin(β)

1/sin θ cos π2 + cos θ sin π2 ? = sec θ

cos π2 = 0

1/sin θ (0) + cos θ sin π2 ? = sec θ

Zero Property of Multiplication

1/0 + cos θ sin π2 ? = sec θ

Identity Property of Addition

1/cos θ sin π2 ? = sec θ

sin π2 = 1

1/cos θ (1) ? = sec θ

Identity Property of Multiplication

1/cos θ ? = sec θ

1/cos θ = sec θ

sec θ = sec 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.

csc ( θ - π/2 ) ? = sec θ

Simplify LHS

sec θ = sec θ ✓

We have verified the identity!

Exercises p.963

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