Question

Hi could you please help me with 1+cot^2x/sec^2x ?

Answer 1

The expression simplifies to `cot^2x`.

Explanation:

First, we need these basic trig identities:

`cscx=1/sinx`

`secx=1/cosx`

`cotx=cosx/sinx`

Now, using the Pythagorean identity,

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

we can derive a new identity if we divide everything by `sin^2x`:

`sin^2x/sin^2x+cos^2x/sin^2x=1/sin^2x`

`1+cot^2x=csc^2x`

Use all these to simplify our expression:

`color(white)=(1+cot^2x)/sec^2x`

`=csc^2x/sec^2x`

`=(1/sin^2x)/(1/cos^2x)`

`=1/sin^2x*cos^2x/1`

`=cos^2x/sin^2x`

`=cot^2x`

That's it. Hope this helped!