The value of $ (1 + \cot^2 heta)(1 + \cos heta)(1 - \cos heta) - (1 - \sin heta)(1 + \sin heta)(1 +

Question

The value of $ (1 + \cot^2 	heta)(1 + \cos 	heta)(1 - \cos 	heta) - (1 - \sin 	heta)(1 + \sin 	heta)(1 + 	an^2 	heta) $ is: ($	heta $ lies in the first quadrant)

Examoogle · Accepted Answer

The correct answer is Option A: 0. Use identity $1 + \cot^2 	heta = \csc^2 	heta$ and $1 - \cos^2 	heta = \sin^2 	heta$ to get $\csc^2 	heta \cdot \sin^2 	heta = 1$. Similarly, $1 - \sin^2 	heta = \cos^2 	heta$ and $1 + 	an^2 	heta = \sec^2 	heta$ gives $\cos^2 	heta \cdot \sec^2 	heta = 1$. The expression simplifies to $1 - 1 = 0$.

Step-by-Step Solution & Concepts:
Shortcut Trick: Use identity $1 + \cot^2 	heta = \csc^2 	heta$ and $1 - \cos^2 	heta = \sin^2 	heta$ to get $\csc^2 	heta \cdot \sin^2 	heta = 1$. Similarly, $1 - \sin^2 	heta = \cos^2 	heta$ and $1 + 	an^2 	heta = \sec^2 	heta$ gives $\cos^2 	heta \cdot \sec^2 	heta = 1$. The expression simplifies to $1 - 1 = 0$.

Formula Used: $$1 + \cot^2 	heta = \csc^2 	heta$$, $$1 + 	an^2 	heta = \sec^2 	heta$$, $$(1 + \cos 	heta)(1 - \cos 	heta) = 1 - \cos^2 	heta = \sin^2 	heta$$, $$(1 - \sin 	heta)(1 + \sin 	heta) = 1 - \sin^2 	heta = \cos^2 	heta$$

Given: Expression involving trigonometric identities with theta in the first quadrant.

Steps: Simplify first part: Using identity $1 + \cot^2 	heta = \csc^2 	heta$ and $(1 + \cos 	heta)(1 - \cos 	heta) = 1 - \cos^2 	heta = \sin^2 	heta$, the first term is $\csc^2 	heta \cdot \sin^2 	heta$.. Simplify second part: Using $(1 - \sin 	heta)(1 + \sin 	heta) = 1 - \sin^2 	heta = \cos^2 	heta$ and $1 + 	an^2 	heta = \sec^2 	heta$, the second term is $\cos^2 	heta \cdot \sec^2 	heta$.. Final calculation: Subtract the two simplified terms: $1 - 1 = 0$.

Answer: A is correct because the entire expression simplifies to $1 - 1$, which equals $0$.

Exam Tip: These identity-based simplifications are common in SSC and Railways exams. Always look for $\sin^2 	heta + \cos^2 	heta = 1$ and reciprocal relationships to quickly cancel out complex terms.

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