Question

use a double - angle formula to find the exact value of the given expression.
1 - 2\sin^{2}15^{circ}
1 - 2\sin^{2}15^{circ}=\square
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Recall the double - angle formula

The double - angle formula for cosine is \(\cos2\alpha=1 - 2\sin^{2}\alpha\).

Step2: Substitute \(\alpha = 15^{\circ}\)

When \(\alpha = 15^{\circ}\), the given expression \(1-2\sin^{2}15^{\circ}\) becomes \(\cos(2\times15^{\circ})\).
So, \(1 - 2\sin^{2}15^{\circ}=\cos30^{\circ}\).

Step3: Find the value of \(\cos30^{\circ}\)

We know that \(\cos30^{\circ}=\frac{\sqrt{3}}{2}\).

Answer:

\(\frac{\sqrt{3}}{2}\)