Exercise 21 Page 883

We want to verify the given trigonometric identity. sec θ csc θ = tan θ + cot θ We will simplify the left-hand side of the equation using the Reciprocal Identities. Then we will simplify the right-hand side of the equation using the Quotient and the Pythagorean Identities to arrive at the left-hand side. Let's start by recalling two of the Reciprocal Identities. sec θ = 1/cos θ, cos θ ≠ 0 [1.0 em] csc θ = 1/sin θ, sin θ ≠ 0We can substitute 1cos θ for sec θ and 1sin θ for csc θ in our expression on the left-hand side. Then we will simplify it.

sec θ csc θ

Substitute

sec θ= 1/cos θ

1/cos θ * csc θ

Substitute

csc θ= 1/sin θ

1/cos θ * 1/sin θ

MultFrac

Multiply fractions

1/cos θ sin θ

Now we can move on to simplifying the expression on the right-hand side. Let's recall the Quotient Identities. tan θ = sin θ/cos θ, cos θ ≠ 0 [1.0 em] cot θ = cos θ/sin θ, sin θ ≠ 0 We can substitute sin θcos θ for tan θ and cos θsin θ for cot θ in our expression. Then we will simplify it.

tan θ + cot θ

Substitute

tan θ= sin θ/cos θ

sin θ/cos θ + cot θ

Substitute

cot θ= cos θ/sin θ

sin θ/cos θ + cos θ/sin θ

▼

Simplify

ExpandFrac

a/b=a * sin θ/b * sin θ

sin θ sin θ/cos θ sin θ + cos θ/sin θ

ExpandFrac

a/b=a * cos θ/b * cos θ

sin θ sin θ/cos θ sin θ + cos θ cos θ/cos θ sin θ

ProdToPowTwoFac

a* a=a^2

sin^2 θ/cos θ sin θ + cos^2 θ/cos θ sin θ

AddFrac

Add fractions

sin^2 θ + cos^2 θ/cos θ sin θ

Finally, let's recall one of the Pythagorean Identities. cos^2 θ + sin ^2 θ = 1 We will substitute 1 for sin ^2 θ + cos^2 θ in our expression.

sin^2 θ + cos^2 θ/cos θ sin θ

Substitute

sin^2 θ + cos^2 θ= 1

1/cos θ sin θ ✓

We obtained the expression on the right-hand side that is equal to simplified form of the expression on the left-hand side. Therefore, we have verified the identity.