The value of tan { cos - 1 ( 1 5 2 ) - sin - 1 ( 4 17 ) } is:

Question

The value of $\tan {\cos^{- 1} (\frac{1}{5 \sqrt{2}}) - \sin^{- 1} (\frac{4}{\sqrt{17}})}$ is:

Smart Achievers · Accepted Answer

(4)

Ans. $\frac{3}{29}$

Explanation:

$\tan {\cos^{- 1} (\frac{1}{5 \sqrt{2}}) - \sin^{- 1} (\frac{4}{\sqrt{17}})}$

Let $A = \cos^{- 1} (\frac{1}{5 \sqrt{2}})$

$∴$ $\cos A = \frac{1}{5 \sqrt{2}}$

$\Rightarrow$ $\cos^{2} A = \frac{1}{50}$

$\Rightarrow$ $\sec^{2} A = 50$

$\Rightarrow$ $\tan^{2} A = 50 - 1 = 49$

and $B = \sin^{- 1} (\frac{4}{\sqrt{17}})$

$\sin B = \frac{4}{\sqrt{17}}$

$\sin^{2} B = \frac{16}{17}$

$c o s e c^{2} B = \frac{17}{16}$

$\cot^{2} B = \frac{17}{16} - 1 = \frac{1}{16}$

$∴$ $\tan A = 7 and \tan B = 4$

Now, $\tan {\cos^{- 1} (\frac{1}{5 \sqrt{2}}) - \sin^{- 1} (\frac{4}{\sqrt{17}})}$

$= \tan (A - B)$

$= \frac{\tan A - \tan B}{1 + \tan A \tan B}$

$= \frac{7 - 4}{1 + 7 \times 4} = \frac{3}{29}$

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