Find the value of cos sin^-1 (-root3/2)

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Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{Let} S=\cos \left\{{\sin }^{-1}\left(-\frac{\sqrt{3}}{2}\right)\right\} \\ \mathrm{Now} \mathrm{We} \mathrm{know} \mathrm{that} {\sin }^{-1}\left(-\frac{\sqrt{3}}{2}\right)=-\frac{\mathrm{\pi }}{3} \\ \Rightarrow S=\cos \left\{-\frac{\mathrm{\pi }}{3}\right\} \\ \mathrm{Also} \cos \left(-\alpha \right)=\cos \left(\alpha \right) \\ \Rightarrow S=\cos \left\{\frac{\mathrm{\pi }}{3}\right\} \\ \Rightarrow S=\frac{1}{2} \\ \mathbf{Hence}\mathbf{ }\mathbf{cos}\left\{{\mathbf{sin}}^{\mathbf{-}\mathbf{1}}\left(\mathbf{-}\frac{\sqrt{\mathbf{3}}}{\mathbf{2}}\right)\right\}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}} \end{gathered}$$

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