Question

How do you simplify the expression `cos^2A(sec^2A-1)`?

Answer 1

`cos^2 A (sec^2 A - 1) = sin^2 A`

with exclusion `A != pi/2 + npi` for integer values of `n`.

Explanation:

Note that:

`sec A = 1/(cos A)`

`sin^2 A + cos^2 A = 1`

So we find:

`cos^2 A (sec^2 A - 1) = (cos^2 A)/(cos^2 A) - cos^2 A`

`color(white)(cos^2 A (sec^2 A - 1)) = 1 - cos^2 A`

`color(white)(cos^2 A (sec^2 A - 1)) = sin^2 A`

Note that this identity does not hold for `A = pi/2 + npi`, when `sec A` is undefined, resulting in the left hand side being undefined but the right hand side defined.