Question

Prove that `tan^2 θ - 1/(cos^2 θ) = -1`.

Answer 1

Given: Let θ be an angle.

To Prove: `tan^2 θ - 1/(cos^2 θ) = -1`.

Proof [Step-wise]:

1. Start with the left-hand side:

LHS = `tan^2 θ - 1/(cos^2 θ)`

2. Recognize:

`1/(cos^2 θ) = sec^2 θ`

So, LHS = tan² θ – sec² θ.

3. Use the Pythagorean identity sec² θ = tan² θ + 1 

Since `sec^2 θ = 1/(cos^2 θ)`

= `(sin^2 θ + cos^2 θ)/(cos^2 θ)` 

= tan² θ + 1

4. Substitute:

LHS = tan² θ – (tan² θ + 1)

= tan² θ – tan² θ – 1

= –1

`tan^2 θ - 1/cos^2 θ = -1`, as required.