<s>[INST] How to Prove Below Inverse Sin How to prove below equation .
$$     \sin^{-1}\frac{3}{5}+\sin^{-1}\frac{8}{17} = \sin^{-1}\frac{77}{85}.$$
I am not able to prove the above equation how can we prove it.
As we know 
$  \sin^{-1} = y$ if $\sin y = x$ where $-1\leq x\leq 1, -\pi/2 \leq y \leq \pi/2.$
 [/INST] Let  $\displaystyle\sin^{-1}\frac35= A$
$\displaystyle\implies (i)\sin A=\frac35$ and $(ii)-\frac\pi2< A<\frac\pi2 $ as the Principal value of sine inverse lies in that range $\displaystyle\implies\cos A>0 \implies \cos A=+\sqrt{1-\left(\frac35\right)^2}=+\frac45$
Similarly for $\displaystyle\sin^{-1}\frac5{13}= B$
Use $\sin(A+B)=\sin A\cos B+\cos A\sin B$
From this, $$\sin^{-1}x+\sin^{-1}y=$$
$$ \begin{cases} \sin^{-1}(x\sqrt{1-y^2}+y\sqrt{1-x^2}) &\mbox{if } x^2+y^2\le1\text{ or if } x^2+y^2>1\text{ and } xy<0  \\ \pi-\sin^{-1}(x\sqrt{1-y^2}+y\sqrt{1-x^2}) & \mbox{if } x^2+y^2>1\text{ and }x,y>0 \\  -\pi-\sin^{-1}(x\sqrt{1-y^2}+y\sqrt{1-x^2}) & \mbox{if } x^2+y^2>1\text{ and }x,y<0 \end{cases} $$
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