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  4. If cos -1((15/17))+2 tan -1((1/5))= cos -1((140/k)) then find k

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Q. If $\cos ^{-1}\left(\frac{15}{17}\right)+2 \tan ^{-1}\left(\frac{1}{5}\right)=\cos ^{-1}\left(\frac{140}{k}\right)$ then find $k$

Inverse Trigonometric Functions

Solution:

$\cos ^{-1} \frac{15}{17}=\tan ^{-1} \frac{8}{15}$
$2 \tan ^{-1}\left(\frac{1}{15}\right)=\tan ^{-1}\left(\frac{5}{12}\right)$
$\therefore \cos ^{-1}\left(\frac{15}{17}\right)+2 \tan ^{-1}\left(\frac{1}{5}\right)$
$=\tan ^{-1}\left(\frac{8}{15}\right)+\tan ^{-1}\left(\frac{5}{12}\right)$
$=\tan ^{-1}\left(\frac{\frac{8}{15}+\frac{5}{12}}{1-\frac{40}{180}}\right)=\tan ^{-1}\left(\frac{171}{140}\right)$
$=\cos ^{-1}\left(\frac{140}{221}\right)$