If $\cos ^{-1}\left(\frac{5}{13}\right)+\cos ^{-1}\left(\frac{3}{5}\right)=\cos ^{-1} x$, then $x$ is equal to

If $\cos ^{-1}\left(\frac{5}{13}\right)+\cos ^{-1}\left(\frac{3}{5}\right)=\cos ^{-1} x$, then $x$ is equal to
  1. $\frac{3}{65}$
  2. $\frac{-36}{65}$
  3. $\frac{-33}{65}$
  4. $-1$

Solution

$\begin{aligned} & \cos ^{-1}\left(\frac{5}{13}\right)+\cos ^{-1}\left(\frac{3}{5}\right)=\cos ^{-1} x \\ & =\cos ^{-1}\left[\frac{5}{13} \cdot \frac{3}{5}-\sqrt{1-\frac{25}{169}} \cdot \sqrt{1-\frac{9}{25}}\right]=\cos ^{-1} x \\ & \Rightarrow \quad \cos ^{-1}\left[\frac{3}{13}-\frac{12}{13} \cdot \frac{4}{5}\right]=\cos ^{-1} x \\ & \Rightarrow \quad \cos ^{-1}\left[\frac{15-48}{65}\right]=\cos ^{-1} x \\ & \therefore \quad x=\frac{-33}{65}\end{aligned}$

Asked in: AP EAMCET 2013

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