The value of the expression tan⁡12cos−1⁡25 is

# The value of the expression $\mathrm{tan}\left(\frac{1}{2}{\mathrm{cos}}^{-1}\frac{2}{\sqrt{5}}\right)$ is

1. A

$2+\sqrt{5}$

2. B

$\sqrt{5}-2$

3. C

$\frac{\sqrt{5}+2}{2}$

4. D

$5+\sqrt{2}$

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### Solution:

$\begin{array}{c}\mathrm{tan}\left(\frac{1}{2}{\mathrm{cos}}^{-1}\frac{2}{\sqrt{5}}\right)=\sqrt{\frac{1-\mathrm{cos}\left[{\mathrm{cos}}^{-1}\left(\frac{2}{\sqrt{5}}\right)\right]}{1+\mathrm{cos}\left[{\mathrm{cos}}^{-1}\left(\frac{2}{\sqrt{5}}\right)\right]}}\\ =\sqrt{\frac{1-\frac{2}{\sqrt{5}}}{1+\frac{2}{\sqrt{5}}}}=\sqrt{\frac{\sqrt{5}-2}{\sqrt{5}+2}×\frac{\sqrt{5}-2}{\sqrt{5}-2}}\\ =\sqrt{\frac{\left(\sqrt{5}-2{\right)}^{2}}{5-4}}=\sqrt{5}-2\end{array}$

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