Q.Prove the following: $\dfrac{\cos(\pi+x)\cos(-x)}{\sin(\pi-x)\cos\left(\dfrac{\pi}{2}+x\right)}=\cot^2x$

Allied angles are angles whose sum or difference with a given angle $\theta$ is an integer multiple of $\dfrac{\pi}{2}$ — i.e. $-\theta,\ \dfrac{\pi}{2}\pm\theta,\ \pi\pm\theta,\ \dfrac{3\pi}{2}\pm\theta,\ 2\pi\pm\theta$. Every trigonometric ratio of an allied angle equals $\pm$ the same or the co-ratio of $\theta$, obtained instantly from the compound-angle formulas of the previous concept. The working rule: (i) write the given angle as $n\cdot90°\pm\theta$ for the smallest possible acute $\theta$; (ii) if $n$ is even, the ratio name stays the same (sin stays sin); if $n$ is odd, the ratio changes to its co-ratio (sin becomes cos, tan becomes cot); (iii) the sign is decided by the quadrant in which …