Exercise 5 Page 882

We want to verify the given trigonometric identity. tan ^2 θ csc ^2 θ = 1 + tan ^2 θ We will simplify the right-hand side of the equation using one of the Pythagorean Identities. Then we will simplify the left-hand side of the equation using the Quotient and the Reciprocal Identities to arrive at the right-hand side. Let's start by recalling one of the Pythagorean Identities. tan ^2 θ +1 =sec^2 θWe can substitute sec^2 θ for 1+tan ^2 θ in our expression on the right-hand side.

1 + tan ^2 θ

Substitute

1+tan ^2 θ = sec^2 θ

sec^2 θ

Now we can move on to simplifying the expression on the left-hand side. Let's recall one of the Quotient Identities. tan θ = sin θ/cos θ, cos θ ≠ 0 We can substitute sin θcos θ for tan θ in our expression.

tan ^2 θ csc ^2 θ

Substitute

tan θ= sin θ/cos θ

( sin θ/cos θ )^2 csc ^2 θ

Next, let's recall one of the Reciprocal Identities. csc θ = 1/sin θ, sin θ ≠ 0 We will substitute 1sin θ for csc θ in our expression. Then we will continue simplifying.

(sin θ/cos θ )^2 csc ^2 θ

Substitute

csc θ= 1/sin θ

(sin θ/cos θ )^2 ( 1/sin θ ) ^2

▼

Simplify

PowQuot

(a/b)^m=a^m/b^m

sin^2 θ/cos^2 θ * 1/sin^2 θ

MultFrac

Multiply fractions

sin^2 θ/cos^2 θ sin^2 θ

ReduceFrac

a/b=.a /sin^2 θ./.b /sin^2 θ.

1/cos^2 θ

a^m/b^m=(a/b)^m

(1/cos θ)^2

Finally, let's recall once again one of the Reciprocal Identities. sec θ = 1/cos θ, cos θ ≠ 0 We can substitute sec θ for 1cos θ in our expression.

(1/cos θ)^2

Substitute

1/cos θ= sec θ

sec ^2 θ ✓

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