Question

Find the exact value of the expression. No decimal answers. Show all work.Hint: Use an identity to simplify the expression.

Answer 1

In trigonometry, we use several identities that help us simplify expressions. One of these is:

`\cos ^2(x)-\sin ^2(x)=\cos (2x)`

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.

`\begin{gathered} \cos ^2((5\pi)/(12))-\sin ^2((5\pi)/(12)) \\ \cos (2\cdot(5\pi)/(12)) \\ \cos ((5\pi)/(6))=-\frac{\sqrt[]{3}}{2} \end{gathered}`

The value of the expression is equal to -(sqrt(3))/2.