use a double - angle formula to find the exact value of the given expression\n2\\cos ^{2}157.5^{\\circ}-1\n2\…

Explanation:

Step1: Recall the double - angle formula

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

Step2: Substitute (\alpha = 157.5^{\circ})

Let (\alpha=157.5^{\circ}), then (2\cos^{2}157.5^{\circ}-1=\cos(2\times157.5^{\circ})).

Step3: Calculate (2\times157.5^{\circ})

(2\times157.5^{\circ}=315^{\circ}).

Step4: Find the value of (\cos(315^{\circ}))

We know that (\cos(315^{\circ})=\cos(360^{\circ}-45^{\circ})). Using the formula (\cos(A - B)=\cos A\cos B+\sin A\sin B) with (A = 360^{\circ}), (B = 45^{\circ}), (\cos(360^{\circ}-45^{\circ})=\cos360^{\circ}\cos45^{\circ}+\sin360^{\circ}\sin45^{\circ}). Since (\cos360^{\circ}=1) and (\sin360^{\circ}=0), then (\cos(360^{\circ}-45^{\circ})=\cos45^{\circ}). And (\cos45^{\circ}=\frac{\sqrt{2}}{2}).

Answer:

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