Question

Question #0042b

Answer 1

The solution is: all `theta` except of `theta = pi/2+pik` with `k` an integer.

Explanation:

`sec^2 theta = 1/cos^2 theta` is not defined for `cos theta = 0`, so it is not defined for `theta = pi/4 +pik` with `k` an integer.

For all other `theta`, we have

`(1-sin^2 theta) * 1/cos^2 theta = cos^2 theta * 1/cos^2 theta = 1`

So, every other `theta` is a solution.

Answer 2

See explanation below.

Explanation:

We have: `(1 - sin^(2)(theta)) sec^(2)(theta)`

One of the Pythagorean identities is `cos^(2)(theta) + sin^(2)(theta) = 1`.

We can rearrange it to get:

`Rightarrow cos^(2)(theta) = 1 - sin^(2)(theta)`

Let's apply this rearranged identity to our proof:

`= cos^(2)(theta) cdot sec^(2)(theta)`

Let's apply the standard trigonometric identity `sec(theta) = frac(1)(cos(theta))`:

`= cos^(2)(theta) cdot frac(1)(cos^(2)(theta))`

`= frac(cos^(2)(theta))(cos^(2)(theta))`

`= 1 " " "` `"Q.E.D."`