Mathematics

Problem

Let

q(x)=\bigl(\cot x\bigr)^{2}

for all

x

where the expression is defined. a. Find

\dfrac{d}{dx}(\cot x)

and express your answer using a single trigonometric function. b. Using your result from part (a) and appropriate differentiation techniques, compute

q'(x)

. c. Show algebraically that

q'(x)=-2\cot x\csc^{2}x

and explain each simplification step clearly.