50tan (3 ^ - 1 ((1)/(2)) + 2 ^ - 1 ( 1 5 )) + 4 2 ((1)/(2) ^ - 1 (2...

Let& S = 50\tan \left(3\tan^{- 1}\dfrac{1}{2} + 2\cos^{- 1}\dfrac{1}{5}\right)+ 4\sqrt{2}\tan \left(\dfrac{1}{2}\tan^{- 1}2\sqrt{2}\right)</p><p>= 50\tan \left(\tan^{- 1}\dfrac{1}{2} +2\left(\tan^{- 1}\dfrac{1}{2} + \tan^{- 1}2\right)\right)+ 4$\sqrt{2}$\tan \left(\dfrac{1}{2}\tan^{- 1}2\sqrt{2}\right)$</p><p>Let~$\dfrac{1}{2}\tan^{- 1}2\sqrt{2} = \alpha ⇒ \tan 2\alpha = 2\sqrt{2}$</p><p>Now~$\dfrac{2tan\alpha}{1 - \tan^{2}\alpha} = 2\sqrt{2}$</p><p>Solving we get,~$tan\alpha = \dfrac{1}{2}$</p><p>So\leftS = 50\tan \left(\tan^{- 1}\dfrac{1}{2} + 2 \cdot \dfrac{π}{2}\right)) + 4\sqrt{2} \times \dfrac{1}{2}$

$= 50\left({tantan}^{- 1}\dfrac{1}{2}\right) + 4$

$= 25 + 4 = 29$