Question

write the following expression as the sine, cosine, or tangent of a double angle. then find the exact value of the expression.
2 cos²(7π/8) - 1
write the following expression as the sine, cosine, or tangent of a double angle. select the correct choice below and in the answer box to complete your choice.
(simplify your answer. type an exact answer, using π as needed. use integers or fractions for any numbers in the expression.)
○ a. 2 cos²(7π/8) - 1 = tan
○ b. 2 cos²(7π/8) - 1 = sin
○ c. 2 cos²(7π/8) - 1 = cos

Explanation:

Step1: Recall the double - angle formula for cosine

The double - angle formula for cosine is \(\cos(2\alpha)=2\cos^{2}\alpha - 1\).
In the given expression \(2\cos^{2}\frac{7\pi}{8}-1\), if we let \(\alpha=\frac{7\pi}{8}\), then by the formula \(2\cos^{2}\alpha - 1=\cos(2\alpha)\).
So \(2\cos^{2}\frac{7\pi}{8}-1=\cos(2\times\frac{7\pi}{8})\).

Step2: Simplify the argument of the cosine function

Calculate \(2\times\frac{7\pi}{8}=\frac{7\pi}{4}\).

Answer:

C. \(2\cos^{2}\frac{7\pi}{8}-1 = \cos\frac{7\pi}{4}\)