Kerala KEAM 5 June 2024 Solved Paper

We are tasked with simplifying the expression: $$(\sec A - \cos A)(\tan A - \cot A).$$Step 1: First, express $\sec A$ and $\cot A$ in terms of $\sin A$ and $\cos A$:- $\sec A = \frac{1}{\cos A}$,- $\cot A = \frac{\cos A}{\sin A}$.Substitute these into the given expression: $$\left( \frac{1}{\cos A} - \cos A \right) \left( \tan A - \frac{\cos A}{\sin A} \right).$$Step 2: Simplify the first part $\frac{1}{\cos A} - \cos A$. We get a common denominator: $$\frac{1}{\cos A} - \cos A = \frac{1 - \cos^2 A}{\cos A}.$$Using the identity $1 - \cos^2 A = \sin^2 A$, this becomes: $$\frac{\sin^2 A}{\cos A}.$$Step 3: Now, simplify the second part $\tan A - \frac{\cos A}{\sin A}$: $$\tan A = \frac{\sin A}{\cos A}, \quad \frac{\cos A}{\sin A} = \cot A.$$Thus, we have: $$\frac{\sin A}{\cos A} - \frac{\cos A}{\sin A}.$$To combine these terms, find a common denominator: $$\frac{\sin^2 A - \cos^2 A}{\sin A \cos A}.$$This can be written as: $$\frac{-(\cos^2 A - \sin^2 A)}{\sin A \cos A} = - \frac{\cos 2A}{\sin A \cos A}.$$Step 4: Now multiply the two parts: $$\frac{\sin^2 A}{\cos A} \times \left( -\frac{\cos^2 A - \sin^2 A}{\sin A \cos A} \right) = - \frac{\sin A (1 - \tan^2 A)}{1}.$$Thus, the simplified expression is: $$- \sin A (1 - \tan^2 A).$$Therefore, the correct answer is option (B).Quick Tip: When simplifying trigonometric expressions, converting to sine and cosine can help simplify terms. Look for common identities such as $1 - \cos^2 A = \sin^2 A$ and factor where possible.