Considering only the principal values of the inverse trigonometric functions, the value of

$\tan (\sin^{- 1} (\frac{3}{5}) - 2 \cos^{- 1} (\frac{2}{\sqrt{5}}))$ is [2024]

Question

Considering only the principal values of the inverse trigonometric functions, the value of

$\tan (\sin^{- 1} (\frac{3}{5}) - 2 \cos^{- 1} (\frac{2}{\sqrt{5}}))$ is [2024]

Smart Achievers · Accepted Answer

(2)

$We have, \tan (\sin^{- 1} (\frac{3}{5}) - 2 \cos^{- 1} (\frac{2}{\sqrt{5}}))$

$Let \sin^{- 1} (\frac{3}{5}) = α, 2 \cos^{- 1} (\frac{2}{\sqrt{5}}) = β \Rightarrow \cos \frac{β}{2} = \frac{2}{\sqrt{5}}$

$∵$ $\sin α = \frac{3}{5} \Rightarrow \tan α = \frac{3}{4}$

$\tan β = \frac{2 \tan \frac{β}{2}}{1 - \tan^{2} \frac{β}{2}} = \frac{2 \times \frac{1}{2}}{1 - \frac{1}{4}} = \frac{4}{3}$

$\Rightarrow \tan (α - β) = \frac{\tan α - \tan β}{1 + \tan α \tan β} = \frac{\frac{3}{4} - \frac{4}{3}}{1 + 1} = - \frac{7}{24}$

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