Prove that: $\dfrac{1 + \cos A}{\sin A} + \dfrac{\sin A}{1 + \cos A} = 2 \csc A$.
Step-by-Step Solution
Key Concept: LHS $= \dfrac{(1+\cos A)^2 + \sin^2 A}{\sin A(1+\cos A)} = \dfrac{1 + 2\cos A + \cos^2 A + \sin^2 A}{\sin A(1+\cos A)} = \dfrac{2 + 2\cos A}{\sin A(1+\cos A)} = \dfrac{2(1+\cos A)}{\sin A(1+\cos A)} = \dfrac{2}{\sin A} = 2 \csc A$.
LHS $= \dfrac{1 + 2\cos A + \cos^2 A + \sin^2 A}{\sin A(1+\cos A)} = \dfrac{2 + 2\cos A}{\sin A(1+\cos A)}$. [1.5 Marks]
$= \dfrac{2(1+\cos A)}{\sin A(1+\cos A)} = \dfrac{2}{\sin A} = 2\csc A = $ RHS. Proved! [1.5 Marks]
---
🎯 Official CBSE Marking Scheme:
Combining fractions over $\sin A(1+\cos A)$: 1.5 Marks
Canceling $(1+\cos A)$ to get $2\csc A$: 1.5 Marks
Correct Answer: