Question

How do you simplify the expression `sec^2theta-tan^2theta+cot^2theta`?

Answer 1

This expression simplifies to `csc^2theta`.

Explanation:

Apply the following identities:

•`sectheta = 1/costheta`
•`tantheta = sintheta/costheta`
•`cottheta = costheta/sintheta`

`=1/cos^2theta - sin^2theta/cos^2theta + cos^2theta/sin^2theta`

`=(1 - sin^2theta)/(cos^2theta) + cos^2theta/sin^2theta`

Apply the pythagorean identity `sin^2x + cos^2x = 1 -> 1 - sin^2x = cos^2x`

`=cos^2theta/cos^2theta + cos^2theta/sin^2theta`

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

`=(sin^2theta + cos^2theta)/sin^2theta`

Apply the identity `sin^2beta + cos^2beta = 1`:

`=1/sin^2theta`

Apply the identity `1/sinalpha = cscalpha`.

`=csc^2theta`

Hopefully this helps!