Prove that:
$\cos ^{-1} \frac{3}{5}+\sin ^{-1} \frac{12}{13}=\sin ^{-1} \frac{56}{65}$
To Prove: $\cos ^{-1} \frac{3}{5}+\sin ^{-1} \frac{12}{13}=\sin ^{-1} \frac{56}{65}$
Formula Used: $\sin ^{-1} x+\sin ^{-1} y=\sin ^{-1}\left(x \times \sqrt{1-y^{2}}+y \times \sqrt{1-x^{2}}\right)$
Proof:
$\mathrm{LHS}=\cos ^{-1} \frac{3}{5}+\sin ^{-1} \frac{12}{13} \ldots$ (1)
Let $\cos \theta=\frac{3}{5}$
Therefore $\theta=\cos ^{-1} \frac{3}{5} \ldots$ (2)
From the figure, $\sin \theta=\frac{4}{5}$
$\Rightarrow \theta=\sin ^{-1} \frac{4}{5} \ldots$ (3)
From $(2)$ and $(3)$,
$\cos ^{-1} \frac{3}{5}=\sin ^{-1} \frac{4}{5}$
Substituting in (1), we get
$\mathrm{LHS}=\sin ^{-1} \frac{4}{5}+\sin ^{-1} \frac{12}{13}$
$=\sin ^{-1}\left(\frac{4}{5} \times \sqrt{1-\left(\frac{12}{13}\right)^{2}}+\frac{12}{13} \times \sqrt{1-\left(\frac{4}{5}\right)^{2}}\right)$
$=\sin ^{-1}\left(\frac{4}{5} \times \sqrt{1-\frac{144}{169}}+\frac{12}{13} \times \sqrt{\left.1-\frac{16}{25}\right)}\right.$
$=\sin ^{-1}\left(\frac{4}{5} \times \sqrt{\frac{25}{169}}+\frac{12}{13} \times \sqrt{\frac{9}{25}}\right)$
$=\sin ^{-1}\left(\frac{4}{5} \times \frac{5}{13}+\frac{12}{13} \times \frac{3}{5}\right)$
$=\sin ^{-1}\left(\frac{20}{65}+\frac{36}{65}\right)$
$=\sin ^{-1} \frac{56}{65}$
$=\mathrm{RHS}$
Therefore, LHS = RHS
Hence proved.
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