The value of

Question:

The value of $\sin ^{-1}\left(\frac{12}{13}\right)-\sin ^{-1}\left(\frac{3}{5}\right)$ is equal to :

  1. (1) $\pi-\sin ^{-1}\left(\frac{63}{65}\right)$

  2. (2) $\frac{\pi}{2}-\sin ^{-1}\left(\frac{56}{65}\right)$

  3. (3) $\frac{\pi}{2}-\cos ^{-1}\left(\frac{9}{65}\right)$

  4. (4) $\pi-\cos ^{-1}\left(\frac{33}{65}\right)$


Correct Option: , 2

Solution:

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

$\left(\because x y \geq 0\right.$ and $\left.x^{2}+y^{2} \leq 1\right)$

$\left[\because \sin ^{-1} x-\sin ^{-1} y=\sin ^{-1}\left\{x \sqrt{1-y^{2}}-y \sqrt{1-x^{2}}\right\}\right]$

$=\sin ^{-1}\left(\frac{-33}{65}\right)=\sin ^{-1}\left(\frac{33}{65}\right)$

$=\cos ^{-1}\left(\frac{56}{65}\right)=\frac{\pi}{2}-\sin ^{-1}\left(\frac{56}{65}\right)$

 

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