Question

How do you verify the identity `(sec^2x-1)cos^2x=sin^2x`?

Answer 1

Simplify the left side of the identity without changing the right side of the identity at all. The left side will simplify to `sin^2x`.

Explanation:

`(sec^2x - 1)cos^2x = sin^2x`
Distribute `cos^2x`:
`sec^2xcos^2x - cos^2x = sin^2x`
Recall that `sec^2x` is defined to be the reciprocal of `cos^2x`, or `1/cos^2x`. Therefore, we now have:
`cos^2x/cos^2x - cos^2x = sin^2x`
This simplifies to:
`1 - cos^2x = sin^2x`
Now recall that `sin^2x + cos^2x = 1`, which can be transformed using the Subtraction Property of Equality to `sin^2x = 1 - cos^2x`. Substituting will then give us:
`sin^2x = sin^2x`
So therefore, the identity has been verified.