If ^ -1 3 4 + ^ -1 5 13 = ^ -1 (k), then k= - Vidyadip

MCQ

If $\cot ^{-1} \frac{3}{4}+\sin ^{-1} \frac{5}{13}=\sin ^{-1}(k)$, then $k=$

✓
$\frac{63}{65}$
B
$\frac{12}{13}$
C
$\frac{65}{68}$
D
$\frac{5}{12}$

✓

Answer

Correct option: A.

$\frac{63}{65}$

(a) : Let $\cot ^{-1} \frac{3}{4}=\theta \Rightarrow \cot \theta=\frac{3}{4}$ and $\sin \theta=\frac{1}{\sqrt{1+\cot ^2 \theta}}=\frac{1}{\sqrt{1+(9 / 16)}}=\frac{4}{5}$ Hence, $\cot ^{-1} \frac{3}{4}+\sin ^{-1} \frac{5}{13}=\sin ^{-1} \frac{4}{5}+\sin ^{-1} \frac{5}{13}$ $=\sin ^{-1}\left[\frac{4}{5} \cdot \sqrt{1-\frac{25}{169}}+\frac{5}{13} \cdot \sqrt{1-\frac{16}{25}}\right]$ $=\sin ^{-1} \frac{63}{65}$

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