simplify to a single trigonometric function then determine the exact value of it. answers dealing with…

Explanation:

Step1: Use the double - angle formula

The double - angle formula for cosine is (\cos(A + B)=\cos A\cos B-\sin A\sin B). When (A = B), we have (\cos(2A)=\cos^{2}A-\sin^{2}A). For the expression (\cos^{2}\frac{\pi}{6}-\sin^{2}\frac{\pi}{6}), let (A=\frac{\pi}{6}). Then (\cos^{2}\frac{\pi}{6}-\sin^{2}\frac{\pi}{6}=\cos(2\times\frac{\pi}{6})).

Step2: Simplify the argument of the cosine function

Simplify (2\times\frac{\pi}{6}). We have (2\times\frac{\pi}{6}=\frac{\pi}{3}). So the trigonometric function is (\cos(\frac{\pi}{3})).

Step3: Find the exact value

We know that (\cos(\frac{\pi}{3})=\frac{1}{2}).

Answer:

trig function (=\cos(\frac{\pi}{3})) exact value (=\frac{1}{2})