The value of the expression tan ($\frac{1}{2}$ cos^-1 2/sqrt(5)) is ______.

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KollegeApply · Accepted Answer

The value of the expression tan ($\frac{1}{2}$ cos^-1 2/sqrt(5)) is sqrt(5) - 2. Explanation: We have, tan ($\frac{1}{2}$ cos^-1 2/sqrt(5)) Let $\theta$ = $\frac{1}{2}$ cos^-1 2/sqrt(5) $\Rightarrow$ 2$\theta$ = cos^-1 2/sqrt(5) $\Rightarrow$ cos 2$\theta$ = 2/sqrt(5) $\Rightarrow$ `(1 - tan^2 theta)/(1 + tan^2 theta) = 2/sqrt(5) ......[because cos 2theta = (1 - tan^2 theta)/(1 + tan^2 theta)] $\Rightarrow$ 2 + 2 tan^2 theta = sqrt(5) - sqrt(5) tan^2 theta $\Rightarrow$ sqrt(5) tan^2 theta + 2 tan^2 theta = sqrt(5) - 2 $\Rightarrow$ (sqrt(5) + 2) tan^2 theta = sqrt(5) - 2 $\Rightarrow$ tan2$\theta$ = ((sqrt(5) - 2)(sqrt(5) - 2))/((sqrt(5) + 2)(sqrt(5) - 2)) $\Rightarrow$ tan2$\theta$ = (sqrt(5) - 2)^2/(5 - 4) $\Rightarrow$ tan2$\theta$ = +- (sqrt(5) - 2) $\Rightarrow$ tan2$\theta$ = sqrt(5) - 2, [-(sqrt(5) - 2) "is not required"]`

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