Hello,
\(1) \int\limits \frac{1}{sin^2(x)} \, dx =-\ cotg x\\\\
2)cosec^4(x)/cotg^2(x)= \frac{1}{sin^4(x)} * \frac{sin^2(x)}{cos^2(x)}= \dfrac{4}{sin^2(2x)} \\\\\\
\int\limits{cosec^4(x)/cotg^2(x)} \, dx \\\\
= \int\limits{ \dfrac{4}{sin^2(2x)}} \.dx\\\\
=-\ 2\ cotg(2x) + C\)
