Question

use the formula for the cosine of the difference of two angles to find the exact value of the following expression.
cos (60° - 45°)

rewrite the expression using a sum or difference formula. choose the correct answer below.
a. sin 60° cos 60° + cos 45° sin 45°
b. sin 60° cos 45° - cos 60° sin 45°
c. cos 60° cos 45° - sin 60° sin 45°
d. cos 60° cos 45° + sin 60° sin 45°

find the exact value of the expression.
cos (60° - 45°) = (simplify your answer. type an exact answer, using radicals as needed. use integers or fractions for any numbers in the expression. rationalize all denominators.)

Explanation:

Step1: Recall the cosine - difference formula

The formula for \(\cos(A - B)=\cos A\cos B+\sin A\sin B\). Here \(A = 60^{\circ}\) and \(B=45^{\circ}\), so \(\cos(60^{\circ}-45^{\circ})=\cos60^{\circ}\cos45^{\circ}+\sin60^{\circ}\sin45^{\circ}\)

Step2: Substitute the values of trigonometric functions

We know that \(\cos60^{\circ}=\frac{1}{2}\), \(\cos45^{\circ}=\frac{\sqrt{2}}{2}\), \(\sin60^{\circ}=\frac{\sqrt{3}}{2}\), and \(\sin45^{\circ}=\frac{\sqrt{2}}{2}\)

Substitute these values into the formula:

$$
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$$

Answer:

\(\frac{\sqrt{2}+\sqrt{6}}{4}\)