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The value of $\tan \{\frac{1}{2} \cos^{- 1} (\frac{\sqrt{5}}{3})\},$ is

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3+52

3+5

12(3−5)

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Correct option is C

Let cos−1⁡53=α. Then, cos⁡α=53, Where 0<α<π2Now, tan⁡α2=1−cos⁡α¯1+cos⁡α⇒ tan⁡α2=1−5/31+5/3 = tan⁡α2=3−53+5=(3−5)29−5=12(3−5)⇒ tan⁡12cos−1⁡53=3−52

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The value of tan⁡12cos−1⁡53, is

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The value of $\tan \{\frac{1}{2} \cos^{- 1} (\frac{\sqrt{5}}{3})\},$ is