which expression is equivalent to \\( \\cos ( \\frac { \\pi } { 12 } ) \\cos ( \\frac { 5 \\pi } { 12 } ) +…

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

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

Answer

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}))