Question

What is `cottheta-sectheta+costheta` in terms of `sintheta`?

Answer 1

`(2(1-sin^2theta-sin^3theta))/(sin2theta)`

Explanation:

Express everything in terms of `sintheta` and `costheta`.

`=costheta/sintheta-1/costheta+costheta`

Get a common denominator.

`=cos^2theta/(costhetasintheta)-sintheta/(costhetasintheta)+(cos^2thetasintheta)/(costhetasintheta)`

Combine.

`=(cos^2theta-sintheta+cos^2thetasintheta)/(costhetasintheta)`

Here, use the identities:

• `cos^2theta=1-sin^2theta`
• `2costhetasintheta=sin2theta=>costhetasintheta=(sin2theta)/2`

`=((1-sin^2theta)-sintheta+(1-sin^2theta)sintheta)/((sin2theta)/2)`

`=(1-sin^2theta-sintheta+sin-sin^3theta)/((sin2theta)/2)`

`=(2(1-sin^2theta-sin^3theta))/(sin2theta)`