Question

How do you express `tan theta - cot theta +sintheta` in terms of `cos theta`?

Answer 1

`=(-cos^3(theta)-2cos^2(theta)+cos(theta)+1)/(cos(theta)sqrt(1-cos^2theta))`

Explanation:

`tan(theta)-cot(theta)+sin(theta)`
We have to write in terms of `cos(theta)`

`color(blue)"Let us start by using the identity"`
`tan(theta) = sin(theta)/cos(theta)` and `cot(theta) = cos(theta)/sin(theta)`

We get

`tan(theta)-cot(theta)+sin(theta)`
`=sin(theta)/cos(theta) - cos(theta)/sin(theta) + sin(theta)`

`color(blue)"In order to simplify we need to use Least Common Denominator for all the fractions"`

`=(sin(theta)sin(theta))/(cos(theta)sin(theta)) -(cos(theta)cos(theta))/(cos(theta)sin(theta)) + (sin(theta)cos(theta)sin(theta))/(cos(theta)sin(theta))`

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

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

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

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

`=(-cos^3(theta)-2cos^2(theta)+cos(theta)+1)/(cos(theta)sqrt(1-cos^2theta))`