The value of ^-11213-^-135 is equal to

Question

The value of $\sin^{-1}\dfrac{12}{13}-\sin^{-1}\dfrac35$ is equal to

Accepted Answer

Using $\sin^{-1}x-\sin^{-1}y=\sin^{-1}\left(x\sqrt{1-y^2}-y\sqrt{1-x^2}\right)$ (valid here since $xy\ge0$ and $x^2+y^2\le1$ conditions are met):

$$\sin^{-1}\frac{12}{13}-\sin^{-1}\frac35=\sin^{-1}\left(\frac{12}{13}\cdot\frac45-\frac35\cdot\frac{5}{13}\right)=\sin^{-1}\left(\frac{48}{65}-\frac{15}{65}\right)=\sin^{-1}\frac{33}{65}.$$

$$=\cos^{-1}\frac{56}{65}=\frac{\pi}{2}-\sin^{-1}\frac{56}{65}.$$

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