Question

How do you verify `(1 + sec^2 x) / (1 + tan^2 x) = 1 + cos^2 x`?

Answer 1

Use the Pythagorean Identity `1+tan^2x=sec^2x`.

Explanation:

Recall the Pythagorean Identity `1+tan^2x=sec^2x` (this can be derived by dividing the identity `sin^2x+cos^2x=1` by `cos^2x`). The key to this problem is applying this identity.

Since `1+tan^2x=sec^2x`, we can replace the `1+tan^2x` in the denominator with `sec^2x`:
`(1+sec^2x)/(sec^2x)=1+cos^2x`

Now we can break the fraction up in two:
`1/sec^2x+sec^2x/sec^2x=1+cos^2x`

Since `1/secx=cosx`, `1/sec^2x=cos^2x`; and `sec^2x/sec^2x=1`. So:
`cos^2x+1=1+cos^2x`

Using the commutative property of addition we can rearrange the left side of the equation to match the right:
`1+cos^2x=1+cos^2x`