Proving the identity

Question

What are the steps and transformations for the expression in proving the following identity:

tanθ(tanθ+cotθ)=sec^2θ

Answer 1

For identity proofs like this, it's usually best to start with the more complex side and then, by using proven (standard) identities, morph it so it becomes the less complicated side.

I will start with tan(θ)(tan(θ) + cot(θ))

Distribute to get:

tan²(θ) + tan(θ)•cotθ)

Using the identity cot(θ) = 1/tan((θ) change the second half of this expression:

tan²(θ) + tan(θ)•(1/tan(θ))

tan²(θ) + 1

Using the Pythagorean Identity: sec²(θ) = tan²(θ) + 1 it becomes

sec²(θ)

Which gives us sec²(θ) = sec²(θ)

Answer 2

tanθ(tanθ+cotθ)=sec^2θ

1. Distribute: tan²θ + tanθ cot θ
2. = tan²θ + 1
3. = sec²θ (because of a version of the pythagorean identity)

cos²θ + sin²θ = 1 : Divide each term by cos²θ

1 + tan²θ = sec²θ