Answer
a) $$-\frac{\cot^22\theta}{4}+C$$
b) $$-\frac{\csc^22\theta}{4}+C$$
Work Step by Step
$$A=\int\csc^22\theta\cot2\theta d\theta$$
a) We set $u=\cot2\theta$.
Then $$du=-2\csc^22\theta d\theta$$
That means, $$\csc^22\theta d\theta=-\frac{1}{2}du$$
Therefore, $$A=-\frac{1}{2}\int udu=-\frac{1}{2}\times\frac{u^2}{2}+C=-\frac{u^2}{4}+C$$ $$A=-\frac{\cot^22\theta}{4}+C$$
b) We set $u=\csc2\theta$.
Then $$du=-2\csc2\theta\cot2\theta d\theta$$
That means, $$\csc2\theta\cot2\theta d\theta=-\frac{1}{2}du$$
Since we can rewrite $A$ into $$A=\int\csc2\theta\times\csc2\theta\cot2\theta d\theta$$
Therefore, with the substitution, $$A=-\frac{1}{2}\int udu=-\frac{1}{2}\times\frac{u^2}{2}+C=-\frac{u^2}{4}+C$$ $$A=-\frac{\csc^22\theta}{4}+C$$
