Question

which expression is equivalent to \\( \cos ( \frac { \pi } { 12 } ) \cos ( \frac { 5 \pi } { 12 } ) + \sin ( \frac { \pi } { 12 } ) \sin ( \frac { 5 \pi } { 12 } ) \\)?
\\( \cos ( - \frac { \pi } { 3 } ) \\)
\\( \sin ( - \frac { \pi } { 3 } ) \\)
\\( \cos ( \frac { \pi } { 2 } ) \\)
\\( \sin ( \frac { \pi } { 2 } ) \\)

Explanation:

Step1: Recall the cosine difference formula

The formula for \(\cos(A - B)=\cos A\cos B+\sin A\sin B\). Let \(A=\frac{\pi}{12}\) and \(B = \frac{5\pi}{12}\), then \(\cos(\frac{\pi}{12})\cos(\frac{5\pi}{12})+\sin(\frac{\pi}{12})\sin(\frac{5\pi}{12})=\cos(\frac{\pi}{12}-\frac{5\pi}{12})\).

Step2: Simplify the expression inside the cosine function

\(\frac{\pi}{12}-\frac{5\pi}{12}=\frac{\pi - 5\pi}{12}=\frac{-4\pi}{12}=-\frac{\pi}{3}\). So \(\cos(\frac{\pi}{12}-\frac{5\pi}{12})=\cos(-\frac{\pi}{3})\). Also, we know that the cosine function is an even function, i.e., \(\cos(-x)=\cos x\), and \(\cos(-\frac{\pi}{3})=\cos(\frac{\pi}{3})=\frac{1}{2}\).

Answer:

\(\cos(-\frac{\pi}{3})\)