Question

If $\cos^{-1} \frac{3}{5} + \cos^{-1} \frac{12}{13} = \cos^{-1} k$, then the value of $k$ is

• A. $\frac{16}{65}$
• B. $\frac{12}{65}$
• C. $\frac{11}{65}$
• D. $\frac{19}{65}$

Answer 1

Answer: (A)

$\frac{16}{65}$

Answer 2

We have,
$\cos^{-1} \frac{3}{5} + \cos^{-1} \frac{12}{13} = \cos^{-1}\,k$
\Rightarrow \cos^{-1}\left\\{\frac{3}{5} \cdot \frac{12}{13}-\sqrt{1-\left(\frac{3}{5}\right)^{2}\sqrt{1-\left(\frac{12}{13}\right)^{2}}}\right\\} = \cos^{-1}\,k
[\because \cos^{-1}\,x + \cos^{-1}\, y = \cos^{-1}\left\\{xy-\sqrt{1-x^{2}}\sqrt{1-y^{2}}\right\\}, if
$-1 \le x, y \le 1$ and $x+y \ge 0]$
\Rightarrow \cos^{-1}\left\\{\frac{36}{65}-\frac{4}{5} \times \frac{5}{13}\right\\} = \cos^{-1}\,k
\Rightarrow \cos^{-1}\left\\{\frac{36}{65}-\frac{20}{65}\right\\} = \cos^{-1}\,k
$\Rightarrow \cos^{-1} \frac{16}{65} = \cos^{-1}\,k$
$\Rightarrow k=\frac{16}{65}$