Solve the following expression:$$\sec^2 A + (\csc^2 A - 1) - (1 + an^2 A) + \cot^2 A = ext{_______}$$

Question

Solve the following expression:$$\sec^2 A + (\csc^2 A - 1) - (1 + 	an^2 A) + \cot^2 A = 	ext{_______}$$

Examoogle · Accepted Answer

The correct answer is Option A: 2\cot^2 A. Use trigonometric identities to simplify: (cosec² A - 1) = cot² A and (1 + tan² A) = sec² A. Substitute these into the original expression to cancel terms.

Step-by-Step Solution & Concepts:
Shortcut Trick: Use trigonometric identities to simplify: (cosec² A - 1) = cot² A and (1 + tan² A) = sec² A. Substitute these into the original expression to cancel terms.

Formula Used: $$\sec^2 A - 	an^2 A = 1, \quad \csc^2 A - \cot^2 A = 1$$

Given: Expression to evaluate: sec² A + (cosec² A - 1) - (1 + tan² A) + cot² A

Steps: Simplify using identities: We know the identities: $\csc^2 A - 1 = \cot^2 A$ and $1 + 	an^2 A = \sec^2 A$. Substitute these into the expression: $\sec^2 A + (\cot^2 A) - (\sec^2 A) + \cot^2 A$.. Group like terms: Group the $\sec^2 A$ terms and the $\cot^2 A$ terms: $(\sec^2 A - \sec^2 A) + (\cot^2 A + \cot^2 A)$.. Calculate final result: The $\sec^2 A$ terms cancel out, leaving $0 + 2\cot^2 A = 2\cot^2 A$.

Answer: A is correct because the expression simplifies to 2cot² A after applying trigonometric identities.

Exam Tip: Trigonometric simplification is a core component of Calculus and Coordinate Geometry. Always look to convert expressions into simpler base terms using fundamental identities like $\sin^2 	heta + \cos^2 	heta = 1$.

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