Problem 36 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the exact value of the expression. $$\cos \frac{\pi}{16} \cos \frac{3 \pi}{16}-\sin \frac{\pi}{16} \sin \frac{3 \pi}{16}$$

Step-by-Step Solution

Answer

The exact value of the expression is \(\frac{\sqrt{2}}{2}\).

Step 1: Identify the Identity

In this step, identify the cosine of a sum of two angles identity: \(\cos(A+B) = \cos A \cos B - \sin A \sin B\). The given expression fits this formula with \(A = \frac{\pi}{16}\) and \(B = \frac{3\pi}{16}\)

Step 2: Apply the Identity

Apply the formula \(\cos(A+B) = \cos A \cos B - \sin A \sin B\). So, \(\cos \frac{\pi}{16} \cos \frac{3 \pi}{16}-\sin \frac{\pi}{16} \sin \frac{3\pi}{16}\) simplifies to \(\cos(\frac{\pi}{16} + \frac{3\pi}{16})\)

Step 3: Simplify the Expression

Simplify \(\cos(\frac{\pi}{16} + \frac{3\pi}{16})\) to \(\cos \frac{4\pi}{16}\), which further simplifies to \(\cos \frac{\pi}{4}\)

Step 4: Calculate the Exact Value

Now calculate the exact value of \(\cos \frac{\pi}{4}\), which is \(\frac{\sqrt{2}}{2}\). Hence, the exact value of the given expression is \(\frac{\sqrt{2}}{2}\)