Kerala KEAM 5 June 2024 Solved Paper

We are asked to simplify the expression $\cos{A} \cos{2A}$. We can use the product-to-sum identity for cosines, which states: $$\cos{A} \cos{B} = \frac{1}{2} [\cos(A - B) + \cos(A + B)]$$Substitute $A = A$ and $B = 2A$ into the identity: $$\cos{A} \cos{2A} = \frac{1}{2} [\cos(A - 2A) + \cos(A + 2A)] = \frac{1}{2} [\cos{-A} + \cos{3A}]$$Since $\cos{-A} = \cos{A}$, this simplifies to: $$\cos{A} \cos{2A} = \frac{1}{2} [\cos{A} + \cos{3A}]$$Next, recall that $\cos{A}$ can be written as $\frac{\sin{4A}}{4 \sin{A}}$, which gives us: $$\cos{A} \cos{2A} = \frac{\sin{4A}}{4 \sin{A}}$$Thus, the correct answer is option (A), $\frac{\sin{4A}}{4 \sin{A}}$. Quick Tip: When simplifying trigonometric expressions involving products, use product-to-sum or sum-to-product identities to break down the expression into simpler terms.