Question

Prove `(sin^2x-tan^2x)/(1-sec^2x) =sin^2x`? The only way I see is to change the `1-sec^2x` to a negative tangent, and cancel out the tangent on top. But, I’m pretty sure you can’t do that, can I?

Answer 1

See explanation...

Explanation:

`(sin^2 x - tan^2 x)/(1-sec^2 x) = (sin^2 x - sin^2 x / cos^2 x)/(1-sec^2 x)`

`color(white)((sin^2 x - tan^2 x)/(1-sec^2 x)) = (sin^2 x(1 - 1 / cos^2 x))/(1-sec^2 x)`

`color(white)((sin^2 x - tan^2 x)/(1-sec^2 x)) = (sin^2 x(1 - sec^2 x))/(1-sec^2 x)`

`color(white)((sin^2 x - tan^2 x)/(1-sec^2 x)) = sin^2 x`

Note that the domains of `tan x` and `sec x` both exclude `x = pi/2 + kpi` for any integer `k`.

In addition note that we require `1-sec^2 x != 0` and hence we also need to exclude `x = kpi` for any integer `k`.

So the identity we found excludes `x = (kpi)/2` for any integer `k`