# Find the exact value of the expression \tan[\cos^{-1}(-\frac{\sqrt{3}}{2})]ZS

Find the exact value of the expression $\mathrm{tan}\left[{\mathrm{cos}}^{-1}\left(-\frac{\sqrt{3}}{2}\right)\right]$ Do not use a calculator.
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Consider the given inverse trigonometry $\mathrm{tan}\left[{\mathrm{cos}}^{-1}\left(-\frac{\sqrt{3}}{2}\right)\right]$
As we know that
${\mathrm{cos}}^{-1}\frac{\sqrt{3}}{2}=\frac{5\pi }{6}$
Then
$\mathrm{tan}\left[{\mathrm{cos}}^{-1}\left(-\frac{\sqrt{3}}{2}\right)\right]=\mathrm{tan}\left[{\mathrm{cos}}^{-1}\mathrm{cos}\frac{5x}{6}\right]$
Use the formula ${\mathrm{cos}}^{-1}\mathrm{cos}\theta =\theta$
$\mathrm{tan}\left[{\mathrm{cos}}^{-1}\left(-\frac{\sqrt{3}}{2}\right)\right]=\mathrm{tan}\left[\frac{5\pi }{6}\right]$
$=-\frac{1}{\sqrt{3}}$
Jeffrey Jordon