Question

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

Answer 1

`[(1 - sin(x))^(3/2)sqrt(1 + sin(x))]/(sin(x))`

Explanation:

We first have to put everything at the same denominator.

`cos(x)/sin(x) - cos(x) = (cos(x) - sin(x).cos(x))/(sin(x)) = [(cos(x))(1 - sin(x))]/(sin(x))`

We know that :
`cos(x) = sqrt(1 - sin^2(x)) = sqrt(1 - sin(x))sqrt(1 + sin(x)).`

Therefor,
`cot(x) - cos(x) = [(1 - sin(x))^(3/2)sqrt(1 + sin(x))]/(sin(x))`