Question

simplify without a calculator:
1 - 2\sin^{2}(15^{\circ})

Explanation:

Step1: Use double - angle formula

The double - angle formula for cosine is \(\cos(2\alpha)=1 - 2\sin^{2}\alpha\). Here \(\alpha = 15^{\circ}\), so \(1-2\sin^{2}(15^{\circ})=\cos(2\times15^{\circ})\).

Step2: Calculate the value

\(2\times15^{\circ}=30^{\circ}\), and \(\cos(30^{\circ})=\frac{\sqrt{3}}{2}\).

Answer:

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