Prove that

${\sin }^{-1}\frac{4}{5}+{\cos }^{-1}\frac{2}{\sqrt{5}}={\cot }^{-1}\frac{2}{11}$.

Prove that

${\sin }^{-1}\frac{3}{5}+{\sin }^{-1}\frac{8}{17}={\cos }^{-1}\frac{36}{85}$.

Prove that

${\sin }^{-1}\frac{8}{17}+{\sin }^{-1}\frac{3}{5}={\tan }^{-1}\frac{77}{36}$.

Prove that

${\sin }^{-1}\frac{x}{\sqrt{1+x^2}}+{\cos }^{-1}\frac{x+1}{\sqrt{x^2+2x+2}}={\tan }^{-1}\left(x^2+x+1\right)$.