In trigonometry, we use several identities that help us simplify expressions. One of these is:
[tex]\cos ^2(x)-\sin ^2(x)=\cos (2x)[/tex]
If we look closely, and replace x by the argument of our question, we will see that the expression from our question is already on this form, so we can use the identity above to rewrite the expression.
[tex]\begin{gathered} \cos ^2(\frac{5\pi}{12})-\sin ^2(\frac{5\pi}{12}) \\ \cos (2\cdot\frac{5\pi}{12}) \\ \cos (\frac{5\pi}{6})=-\frac{\sqrt[]{3}}{2} \end{gathered}[/tex]
The value of the expression is equal to -(sqrt(3))/2.
