Find the value of the expression ( ^ - 1 3 5 + ^ - 1 3 2 ). - Vidyadip

Question

Find the value of the expression $\tan \left( {{{\sin }^{ - 1}}\frac{3}{5} + {{\cot }^{ - 1}}\frac{3}{2}} \right)$.

✓

Answer

Putting, ${\sin ^{ - 1}}\frac{3}{5} = x$ and ${\cot ^{ - 1}}\frac{3}{2} = y$ so that $\sin x = \frac{3}{5}$ and $\cot y = \frac{3}{2}$
Now, $\cos x = \sqrt {1 - {{\sin }^2}x}$ $= \sqrt {1 - \frac{9}{{25}}} = \sqrt {\frac{{16}}{{25}}} = \frac{4}{5}$
And $\tan x = \frac{{\sin x}}{{\cos x}} = \frac{3}{4}$
and $\tan y = \frac{1}{{\cot y}} = \frac{2}{3}$
$\therefore \tan \left( {{{\sin }^{ - 1}}\frac{3}{5} + {{\cot }^{ - 1}}\frac{3}{2}} \right)$
= tan(x + y)
$ = \frac{{\tan x + \tan y}}{{1 - \tan x\tan y}} = \frac{{\frac{3}{4} + \frac{2}{3}}}{{1 - \frac{3}{4} \times \frac{2}{3}}}$
$ = \frac{{\frac{{17}}{{12}}}}{{\frac{1}{2}}} = \frac{{17}}{6}$

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