May/June 2026 Paper 31

By differentiating $\frac{\cos 4x}{\sin 4x}$, show that if $y=\cot 4x$, then $\frac{\mathrm{d}y}{\mathrm{d}x}=-4{\mathrm{cosec}}^{2}4x$.

The variables $x$ and $y$ satisfy the differential equation

It is given that $y=\frac{1}{16}\pi$ when $x=-\frac{2}{3}$.

Solve the differential equation, and find the value of $y$ in the range $-\frac{1}{8}\pi \le y\le 0$ when $x=1$.