Prove that: $\dfrac{\cos A}{1 + \sin A} + \dfrac{1 + \sin A}{\cos A} = 2 \sec A$.
Step-by-Step Solution
Key Concept: LHS $= \dfrac{\cos^2 A + (1 + \sin A)^2}{\cos A (1 + \sin A)} = \dfrac{\cos^2 A + 1 + 2\sin A + \sin^2 A}{\cos A (1 + \sin A)} = \dfrac{2 + 2\sin A}{\cos A (1 + \sin A)} = \dfrac{2(1 + \sin A)}{\cos A (1 + \sin A)} = \dfrac{2}{\cos A} = 2 \sec A$.
LHS $= \dfrac{\cos^2 A + 1 + 2\sin A + \sin^2 A}{\cos A(1 + \sin A)} = \dfrac{2 + 2\sin A}{\cos A(1 + \sin A)}$. [1.0 Mark]
$= \dfrac{2(1 + \sin A)}{\cos A(1 + \sin A)} = \dfrac{2}{\cos A} = 2\sec A = $ RHS. Proved! [1.0 Mark]
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🎯 Official CBSE Marking Scheme:
Combining fractions into $\dfrac{2 + 2\sin A}{\cos A(1 + \sin A)}$: 1.0 Mark
Canceling $(1 + \sin A)$ to get $2\sec A$: 1.0 Mark
Correct Answer: