To find the exact value of [tex]\( \tan 30^\circ \times \cos 30^\circ \)[/tex]:
1. Determine [tex]\( \tan 30^\circ \)[/tex]:
[tex]\[
\tan 30^\circ = \frac{1}{\sqrt{3}}
\][/tex]
2. Determine [tex]\( \cos 30^\circ \)[/tex]:
[tex]\[
\cos 30^\circ = \frac{\sqrt{3}}{2}
\][/tex]
3. Multiply [tex]\( \tan 30^\circ \)[/tex] by [tex]\( \cos 30^\circ \)[/tex]:
[tex]\[
\tan 30^\circ \times \cos 30^\circ = \left(\frac{1}{\sqrt{3}}\right) \times \left(\frac{\sqrt{3}}{2}\right)
\][/tex]
4. Simplify the multiplication:
[tex]\[
\left(\frac{1}{\sqrt{3}}\right) \times \left(\frac{\sqrt{3}}{2}\right) = \frac{1 \times \sqrt{3}}{\sqrt{3} \times 2} = \frac{\sqrt{3}}{2\sqrt{3}} = \frac{1}{2}
\][/tex]
Therefore, the exact value of [tex]\( \tan 30^\circ \times \cos 30^\circ \)[/tex] in its simplest form is:
[tex]\[
\boxed{\frac{1}{2}}
\][/tex]
