Problem 12 - Precalculus Essentials

Question

Verify each identity. $$\cos \left(x-\frac{5 \pi}{4}\right)=-\frac{\sqrt{2}}{2}(\cos x+\sin x)$$

Step-by-Step Solution

Answer

The given expression \(\cos \left(x-\frac{5 \pi}{4}\right)=-\frac{\sqrt{2}}{2}(\cos x+\sin x)\) is indeed a valid trigonometric identity as per the verification process involving usage of the formula for cosine difference and simplification.

Step 1: Recall the formula for cosine of difference of two angles

The formula for cosine of difference of two angles (cos (A-B)) is given by:\[cos(A-B) = cosA cosB + sinA sinB\]

Step 2: Apply the potential trigonometric identity

Start with the left side of the given potential trigonometric identity, \[\cos \left(x-\frac{5 \pi}{4}\right)\] and apply the formula from step 1.\[\cos \left(x-\frac{5 \pi}{4}\right) = \cos x \cos \left(\frac{5 \pi}{4}\right) + \sin x \sin \left(\frac{5 \pi}{4}\right)\]

Step 3: Evaluate the trigonometric functions

The cos(5π/4) and sin(5π/4) are each equal to -√2/2. Substitute these values in the equation obtained in step 2.\[\cos \left(x-\frac{5 \pi}{4}\right) = \cos x \left(-\frac{\sqrt{2}}{2}\right) + \sin x \left(-\frac{\sqrt{2}}{2}\right)\]

Step 4: Simplify the obtained expression

Now simplify the expression obtained in step 3 by factoring out -√2/2.\[\cos (x - \frac{5 \pi}{4}) = -\frac{\sqrt{2}}{2} (\cos x + \sin x)\] which is the right side of the identity. This shows that the given formula is an identity, thus completing the verification process.