Prove that
Question:

Prove that tan-1 1/4 + tan-1 2/9 = sin-1 1/√5

Solution:

Taking the LHS,

tan-1 1/4 + tan-1 2/9

$=\tan ^{-1} \frac{\frac{1}{4}+\frac{2}{9}}{1-\frac{1}{4} \cdot \frac{2}{9}}=\tan ^{-1} \frac{9+8}{36-2}=\tan ^{-1} \frac{1}{2}=\sin ^{-1} \frac{1}{\sqrt{5}}$

= RHS

– Hence Proved