Kerala KEAM 5 June 2024 Solved Paper

We are tasked with simplifying the expression: $$\frac{\sin 7x + \sin 5x}{\cos 7x + \cos 5x}.$$Step 1: Use the sum-to-product identities for sine and cosine. The sum-to-product identity for sine is: $$\sin A + \sin B = 2 \sin\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right),$$and for cosine: $$\cos A + \cos B = 2 \cos\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right).$$Step 2: Apply the sum-to-product identities to the given expression:- For $\sin 7x + \sin 5x$, we have: $$\sin 7x + \sin 5x = 2 \sin\left(\frac{7x + 5x}{2}\right) \cos\left(\frac{7x - 5x}{2}\right) = 2 \sin(6x) \cos(x).$$- For $\cos 7x + \cos 5x$, we have: $$\cos 7x + \cos 5x = 2 \cos\left(\frac{7x + 5x}{2}\right) \cos\left(\frac{7x - 5x}{2}\right) = 2 \cos(6x) \cos(x).$$Step 3: Substitute these into the original expression: $$\frac{\sin 7x + \sin 5x}{\cos 7x + \cos 5x} = \frac{2 \sin(6x) \cos(x)}{2 \cos(6x) \cos(x)}.$$Simplify by canceling out $2 \cos(x)$: $$\frac{\sin(6x)}{\cos(6x)} = \tan(6x).$$Thus, the simplified form of the expression is $\tan 6x$.Therefore, the correct answer is option (E).Quick Tip: When simplifying trigonometric expressions, apply sum-to-product identities to break down complex terms. This can often lead to a simpler form for easier calculation or understanding.