Kerala KEAM 5 June 2024 Solved Paper

Step 1: We are given the expression: $$\left( 1 + \cos \frac{\pi}{8} \right) \left( 1 + \cos \frac{7\pi}{8} \right).$$We can simplify the second cosine term using the identity $\cos \left( \pi - x \right) = -\cos x$: $$\cos \frac{7\pi}{8} = \cos \left( \pi - \frac{\pi}{8} \right) = - \cos \frac{\pi}{8}.$$Thus, the expression becomes: $$\left( 1 + \cos \frac{\pi}{8} \right) \left( 1 - \cos \frac{\pi}{8} \right).$$Step 2: Recognizing this as a difference of squares, we can simplify it as: $$1^2 - \left( \cos \frac{\pi}{8} \right)^2 = 1 - \cos^2 \frac{\pi}{8}.$$Using the Pythagorean identity $\sin^2 x + \cos^2 x = 1$, we can write: $$1 - \cos^2 \frac{\pi}{8} = \sin^2 \frac{\pi}{8}.$$Step 3: Thus, the given expression simplifies to: $$\sin^2 \frac{\pi}{8}.$$Step 4: Now, we use the known value for $\sin \frac{\pi}{8}$, which is $\frac{\sqrt{2 - \sqrt{2}}}{2}$. Therefore, we can write: $$\sin^2 \frac{\pi}{8} = \left( \frac{\sqrt{2 - \sqrt{2}}}{2} \right)^2 = \frac{2 - \sqrt{2}}{4}.$$Step 5: Simplifying this, we get: $$\frac{1}{2\sqrt{2}} (\sqrt{2} - 1).$$Thus, the correct answer is option (C). Quick Tip: Use the identity $\cos(\pi - x) = -\cos(x)$ to simplify trigonometric expressions involving angles that sum to $\pi$. Also, recognize the difference of squares to easily simplify products of trigonometric terms.