Prove that
Question:

Prove that $\cos ^{-1} \frac{12}{13}+\sin ^{-1} \frac{3}{5}=\sin ^{-1} \frac{56}{65}$

Solution:

$\cos ^{-1} \frac{12}{13}$

$=\sin ^{-1} \sqrt{1-\left(\frac{12}{13}\right)^{2}}$

$=\sin ^{-1} \sqrt{1-\frac{144}{169}}$                 $\left(\cos ^{-1} x=\sin ^{-1} \sqrt{1-x^{2}}\right)$

$=\sin ^{-1} \sqrt{\frac{25}{169}}$

$=\sin ^{-1} \frac{5}{13}$

$\therefore \cos ^{-1} \frac{12}{13}+\sin ^{-1} \frac{3}{5}$

$=\sin ^{-1} \frac{5}{13}+\sin ^{-1} \frac{3}{5}$

$=\sin ^{-1}\left(\frac{5}{13} \times \sqrt{1-\left(\frac{3}{5}\right)^{2}}+\frac{3}{5} \times \sqrt{1-\left(\frac{5}{13}\right)^{2}}\right)$                $\left[\sin ^{-1} x+\sin ^{-1} y=\sin ^{-1}\left(x \sqrt{1-y^{2}}+y \sqrt{1-x^{2}}\right)\right]$

$=\sin ^{-1}\left(\frac{5}{13} \times \sqrt{1-\frac{9}{25}}+\frac{3}{5} \times \sqrt{1-\frac{25}{169}}\right)$

$=\sin ^{-1}\left(\frac{5}{13} \times \sqrt{\frac{16}{25}}+\frac{3}{5} \times \sqrt{\frac{144}{169}}\right)$

$=\sin ^{-1}\left(\frac{5}{13} \times \frac{4}{5}+\frac{3}{5} \times \frac{12}{13}\right)$

$=\sin ^{-1}\left(\frac{20}{65}+\frac{36}{65}\right)$

$=\sin ^{-1} \frac{56}{65}$