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Introduction to Trigonometry and Its Applications
NCERT Exemplar Ch 08
CBSE_NCERT_EXEMPLAR_CH08
Grade 10

Question:

Prove that: $\dfrac{\cos A}{1 + \sin A} + \dfrac{1 + \sin A}{\cos A} = 2 \sec A$.

Step-by-Step Solution

Key Concept: Take common denominator $\cos A (1 + \sin A)$ and use $\sin^2 A + \cos^2 A = 1$.
Stepwise Solution:

\text{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)}. [1.0 Mark]

Since $\cos^2 A + \sin^2 A = 1$: $\text{Numerator} = 1 + 1 + 2 \sin A = 2 + 2 \sin A = 2(1 + \sin A)$. [1.0 Mark]

\text{LHS} = \dfrac{2(1 + \sin A)}{\cos A (1 + \sin A)} = \dfrac{2}{\cos A} = 2 \sec A = \text{RHS}$. Proved! [1.0 Mark]

Marking Scheme:

• Common denominator and expansion: 1.0 Mark
• Applying identity $\sin^2 A + \cos^2 A = 1$: 1.0 Mark
• Cancelling $(1+\sin A)$ to get $2\sec A$: 1.0 Mark

Correct Answer:
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