Calculus question

Question

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

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

Answer 1

Recall tan(x/2) = [1-cos(x)]/sin(x) and cot(x/2) = [1+cos(x)]/sin(x)

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

= [1-cos(x)]/sin(x) - [1+cos(x)]/sin(x)

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

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

= -2cot(x)

Recall that the derivative of cot(x) is -csc²(x), so:

f(x) = -2cot(x)

f'(x) = 2csc²(x)

Hope this helped!

Answer 2

f '(x)= (sec²(x/2)+csc²(x/2))/2

= (1/2)(1/cos²(x/2) + 1/sin²(x/2)) = (1/2)((sin²(x/2)+cos²(x/2))/(sin²(x/2)cos²(x/2))

= (2/4)(1/(sin²(x/2)cos²(x/2)) = 2/(2sin(x/2)cos(x/2))² = 2/sin²(x)) = 2csc²(x)