sin ^{-1} frac{4}{5} + 2 tan ^{-1} frac{1}{3} | vedclass

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Question

(C) We are given the expression: $\sin ^{-1} \frac{4}{5} + 2 	an ^{-1} \frac{1}{3}$.First, use the formula $2 	an ^{-1} x = 	an ^{-1} \frac{2x}{1-x

Vedclass · Accepted Answer

$\frac{\pi}{2}$

Vedclass · Answer

(C) We are given the expression: $\sin ^{-1} \frac{4}{5} + 2 	an ^{-1} \frac{1}{3}$.First, use the formula $2 	an ^{-1} x = 	an ^{-1} \frac{2x}{1-x^2}$:$2 	an ^{-1} \frac{1}{3} = 	an ^{-1} \frac{2(1/3)}{1-(1/3)^2} = 	an ^{-1} \frac{2/3}{1-1/9} = 	an ^{-1} \frac{2/3}{8/9} = 	an ^{-1} \left(\frac{2}{3} 	imes \frac{9}{8}ight) = 	an ^{-1} \frac{3}{4}$.Now, the expression becomes $\sin ^{-1} \frac{4}{5} + 	an ^{-1} \frac{3}{4}$.Since $	an ^{-1} \frac{3}{4} = 	heta$, then $	an 	heta = \frac{3}{4}$, which implies $\sin 	heta = \frac{3}{5}$ and $\cos 	heta = \frac{4}{5}$.Thus, $	an ^{-1} \frac{3}{4} = \cos ^{-1} \frac{4}{5}$.Substituting this back, we get $\sin ^{-1} \frac{4}{5} + \cos ^{-1} \frac{4}{5}$.Using the identity $\sin ^{-1} x + \cos ^{-1} x = \frac{\pi}{2}$, we get the result $\frac{\pi}{2}$.

$\sin ^{-1} \frac{4}{5} + 2 \tan ^{-1} \frac{1}{3}$ is equal to

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