Question

How would you write each expression in terms of sin θ and cos θ and simplify?

`tan^2 θ (1 + cot^2 θ)`

Answer 1

`1/cos^2theta`

Explanation:

Recall that

`tantheta=sintheta/costheta`

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

Furthermore, recall the Pythagorean identity

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

And that `csc^2theta=1/sin^2theta`

So, we have

`cancel(sin^2theta)/cos^2theta*1/cancel(sin^2theta)`

Leaving us with

`1/cos^2theta` if we're staying in terms of sine and cosine.