Solve $3{\tan }^{-1}x+{\cot }^{-1}x=\mathrm{\pi }$

Solve for $x:$

${\tan }^{-1}\left(\frac{x-1}{x-2}\right)+{\tan }^{-1}\left(\frac{x+1}{x+2}\right)=\frac{\mathrm{\pi }}{4}$

Prove that: ${\tan }^2\left({\sec }^{-1}\left(2\right)\right)+{\cot }^2\left({cosec}^{-1}\left(3\right)\right)=11$.

If ${\cos }^{-1}\left(\frac{x}{2}\right)+{\cos }^{-1}\left(\frac{y}{3}\right)=\theta$, then prove that: $9x^2-12xy\cos \theta +4y^2=36{\sin }^2\theta$.