use a double - angle formula to find the exact value of the given expression. 1 - 2 sin²165°. 1 - 2 sin²165°…

Explanation:

Step1: Recall double - angle formula

The double - angle formula for cosine is $\cos(2\alpha)=1 - 2\sin^{2}\alpha$.

Step2: Identify $\alpha$

In the expression $1 - 2\sin^{2}165^{\circ}$, we have $\alpha = 165^{\circ}$. So, $1 - 2\sin^{2}165^{\circ}=\cos(2\times165^{\circ})$.

Step3: Calculate the angle

$2\times165^{\circ}=330^{\circ}$.

Step4: Find the cosine value

We know that $\cos(330^{\circ})=\cos(360^{\circ}- 30^{\circ})$. Since $\cos(A - B)=\cos A\cos B+\sin A\sin B$ and $\cos(360^{\circ}) = 1,\sin(360^{\circ}) = 0$, $\cos(330^{\circ})=\cos(360^{\circ}-30^{\circ})=\cos360^{\circ}\cos30^{\circ}+\sin360^{\circ}\sin30^{\circ}=\cos30^{\circ}=\frac{\sqrt{3}}{2}$.

Answer:

$\frac{\sqrt{3}}{2}$