Problem 11 - Precalculus Essentials

Question

Verify each identity. $$\cos \left(x-\frac{\pi}{4}\right)=\frac{\sqrt{2}}{2}(\cos x+\sin x)$$

Step-by-Step Solution

Answer

The identity \(\cos \left(x-\frac{\pi}{4}\right)=\frac{\sqrt{2}}{2}(\cos x+\sin x)\) is verified as true using the formula for cosine of difference of two angles and subsequent simplification.

Step 1: Apply the cosine of difference formula

First, the formula for the cosine of the difference of two angles is applied to the left-hand side of the equation. The formula is: \(\cos(a - b) = \cos a \cos b + \sin a \sin b\). By applying this formula, the expression \(\cos \left(x-\frac{\pi}{4}\right)\) becomes \(\cos x \cos \left(\frac{\pi}{4}\right) + \sin x \sin \left(\frac{\pi}{4}\right)\).

Step 2: Substituting the values

The values can be substituted for \(\cos \left(\frac{\pi}{4}\right)\) and \(\sin \left(\frac{\pi}{4}\right)\), which are both equal to \(\frac{\sqrt{2}}{2}\). This results in \(\frac{\sqrt{2}}{2} \cos x + \frac{\sqrt{2}}{2} \sin x\).

Step 3: Simplifying the right-hand side

The right-hand side of the equation can be simplified by factoring out the common factor of \(\frac{\sqrt{2}}{2}\). This results in \(\frac{\sqrt{2}}{2} (\cos x + \sin x)\). This matches the given identity so it verifies that the equation holds true.