Question

use an identity to find the exact value of the expression.
cos(45° - 60°)
cos(45° - 60°) =
(type an exact answer, using radicals as needed. rationalize all denominators.)

Explanation:

Step1: Apply cosine difference identity

Recall the identity $\cos(A-B) = \cos A \cos B + \sin A \sin B$. For $A=45^\circ$, $B=60^\circ$:
$\cos(45^\circ-60^\circ) = \cos45^\circ\cos60^\circ + \sin45^\circ\sin60^\circ$

Step2: Substitute known trig values

Use $\cos45^\circ=\sin45^\circ=\frac{\sqrt{2}}{2}$, $\cos60^\circ=\frac{1}{2}$, $\sin60^\circ=\frac{\sqrt{3}}{2}$:
$\cos(45^\circ-60^\circ) = \frac{\sqrt{2}}{2} \cdot \frac{1}{2} + \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2}$

Step3: Calculate each product

Multiply the fractions:
$\cos(45^\circ-60^\circ) = \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}$

Step4: Combine the terms

Add the fractions over common denominator:
$\cos(45^\circ-60^\circ) = \frac{\sqrt{2}+\sqrt{6}}{4}$

Answer:

$\frac{\sqrt{2}+\sqrt{6}}{4}$