Calculus question

Question

Let f(x)=tan(x/2)-cot (x/2) .proof f'(x)=2csc^2x.

Where sin^2x+cos^2x=1,2sin(x/2)cos(x/2)

Answer 1

f(x)=tan(x/2)-cot(x/2)

=[sin(x/2)/cos(x/2)]-(cos(x/2)/sin(x/2)]

=[sin²(x/2)-cos²(x/2)]/{sin(x/2)cos(x/2),,,,,use double angle formulas for numerator and denominator.

=cos(x)/[(sin(x))/2]

=2cot(x)

f'(x)=2csc²(x)