Question

How do you prove that `tan(x/2)-cot(x/2) = -2cotx`?

Answer 1

`tan(x/2)−cot(x/2)=-2cotx` For proof see below.

Explanation:

`tan(x/2)−cot(x/2)`

= `sin(x/2)/cos(x/2)-cos(x/2)/sin(x/2)`

=`(sin^2(x/2)-cos^2(x/2))/(sin(x/2)cos(x/2))`

As `cosx=cos^2(x/2)-sin^2(x/2)` and `sinx=2sin(x/2)cos(x/2)`

Above can be written as `-cosx/(sinx/2)` or

`-2cotx`