Given that
[tex]\cos\theta=\frac{\sqrt{3}}{2}[/tex]we can determinate the sine of this angle using the following identity
[tex]\sin^2\theta+\cos^2\theta=1[/tex]If we substitute the value of the cosine on this identity, we're going to have:
[tex]\begin{gathered} \sin^2\theta+(\frac{\sqrt{3}}{2})^2=1 \\ \sin^2\theta+\frac{3}{4}=1 \\ \sin^2\theta=\frac{1}{4} \\ \sin\theta=\pm\frac{1}{2} \end{gathered}[/tex]The definitions of secant, tangent, and cosecant in terms of the sine and cosine are given by:
[tex]\begin{gathered} \tan\theta=\frac{\sin\theta}{\cos\theta} \\ \sec\theta=\frac{1}{\cos\theta} \\ \csc\theta=\frac{1}{\sin\theta} \end{gathered}[/tex]Using the known values for the sine and cosine functions on those definitions, we have:
[tex]\begin{gathered} \tan\theta=\frac{\pm\frac{1}{2}}{\frac{\sqrt{3}}{2}}=\pm\frac{1}{\sqrt{3}}=\pm\frac{\sqrt{3}}{3}\ne-\sqrt{3} \\ \\ \csc\theta=\frac{1}{\pm\frac{1}{2}}=\pm2\ne\frac{1}{2} \\ \\ \sec\theta=\frac{1}{\frac{\sqrt{3}}{2}}=\frac{2}{\sqrt{3}}=\frac{2\sqrt{3}}{3}\ne-2 \\ \\ \sin\theta=\pm\frac{1}{2}\ne\frac{\sqrt{2}}{2} \end{gathered}[/tex]All options are false.
