If = ^ - 1 , ( 3 5 ), = ^ - 1 , ( 1 3 ), where 0 < , < 2 , then -… - Vidyadip

MCQ

If $\alpha = {\cos ^{ - 1}}\,\left( {\frac{3}{5}} \right),\beta = {\tan ^{ - 1}}\,\left( {\frac{1}{3}} \right)$, where $0 < \alpha ,\beta < \frac{\pi }{2}$, then $\alpha - \beta $ is equal to

✓
${\sin ^{ - 1}}\,\left( {\frac{9}{{5\sqrt {10} }}} \right)$
B
${\cos ^{ - 1}}\,\left( {\frac{9}{{5\sqrt {10} }}} \right)$
C
${\tan ^{ - 1}}\,\left( {\frac{9}{{5\sqrt {10} }}} \right)$
D
${\tan ^{ - 1}}\,\left( {\frac{9}{{14}}} \right)$

✓

Answer

Correct option: A.

${\sin ^{ - 1}}\,\left( {\frac{9}{{5\sqrt {10} }}} \right)$

a
$\cos \alpha = \frac{3}{5},\tan \beta \frac{1}{3}$

$ \Rightarrow \tan \alpha = \frac{4}{3}$

$ \Rightarrow \tan \left( {\alpha - \beta } \right) = \frac{{\frac{4}{3} - \frac{1}{3}}}{{1 + \frac{4}{3},\frac{1}{3}}} = \frac{9}{{13}}$

$ \Rightarrow \sin \left( {\alpha - \beta } \right) = \frac{9}{{5\sqrt {10} }}$

$ \Rightarrow \alpha - \beta = {\sin ^{ - 1}}\left( {\frac{9}{{5\sqrt {10} }}} \right)$

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