Question

find an algebraic expression equivalent to the given expression. hint: form a right triangle.

cos(2cos^{-1}x)

cos(2cos^{-1}x)=\square
(simplify your answer. type an exact answer, using radicals as needed. do not rationalize the denominator.)

Explanation:

Step1: Let \(\theta=\cos^{-1}x\)

By the definition of inverse cosine function, \(\cos\theta = x\), where \(0\leqslant\theta\leqslant\pi\).

Step2: Use the double - angle formula for cosine

The double - angle formula for cosine is \(\cos(2\theta)=2\cos^{2}\theta - 1\).
Since \(\theta = \cos^{-1}x\), substituting \(\cos\theta=x\) into the double - angle formula, we get \(\cos(2\cos^{-1}x)=2x^{2}-1\).

Answer:

\(2x^{2}-1\)