Solve the following trigonometric expression: $\sec^2 A + (\c... | Mathematics MCQ | Examoogle

MathematicsPrevious Year Paper 2024

Solve the following trigonometric expression: $\sec^2 A + (\csc^2 A - 1) - (1 + \tan^2 A) - \cot^2 A = \text{\_\_\_\_\_}$

sec² A

cot² A

Correct Answer

📐 MA • Math Direct FormulaMathematicsPrevious Year Paper 2024

Option C

Quick Summary:

Use the basic identities $\sec^2 A - \tan^2 A = 1$ and $\csc^2 A - \cot^2 A = 1$ . The expression simplifies to $(1) + (\cot^2 A) - (1 + \tan^2 A) - \cot^2 A$ , which reduces quickly.

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📐MAMath SolutionDirect Formula

📋 Given

An algebraic expression containing trigonometric functions: $\sec^2 A + (\csc^2 A - 1) - (1 + \tan^2 A) - \cot^2 A$

🔢 Formula Used

sec⁡2A=1+tan⁡2A,csc⁡2A=1+cot⁡2A\sec^2 A = 1 + \tan^2 A, \quad \csc^2 A = 1 + \cot^2 Asec2A=1+tan2A,csc2A=1+cot2A

⚡ Exam Hall Shortcut / Speed Trick

Use the basic identities $\sec^2 A - \tan^2 A = 1$ and $\csc^2 A - \cot^2 A = 1$ . The expression simplifies to $(1) + (\cot^2 A) - (1 + \tan^2 A) - \cot^2 A$ , which reduces quickly.

⚠️ Common Student Trap / Pitfall

A common mistake is incorrectly distributing the negative sign across the terms inside the parentheses or forgetting that $\csc^2 A - 1 = \cot^2 A$ .

📊 Diagram / Illustration

🔢 Step-by-Step Solution

Simplify using identities

We know that $\csc^2 A - 1 = \cot^2 A$ and $1 + \tan^2 A = \sec^2 A$ .

csc⁡2A−1=cot⁡2A,1+tan⁡2A=sec⁡2A\csc^2 A - 1 = \cot^2 A, \quad 1 + \tan^2 A = \sec^2 Acsc2A−1=cot2A,1+tan2A=sec2A

Substitute and rearrange

Substitute these identities into the original expression: $\sec^2 A + \cot^2 A - \sec^2 A - \cot^2 A$ .

sec⁡2A+cot⁡2A−sec⁡2A−cot⁡2A\sec^2 A + \cot^2 A - \sec^2 A - \cot^2 Asec2A+cot2A−sec2A−cot2A

Final cancellation

Group the terms to see that they cancel out: $(\sec^2 A - \sec^2 A) + (\cot^2 A - \cot^2 A) = 0 + 0 = 0$ .

000

✅

C is correct because after simplifying the expression using fundamental trigonometric identities, all terms cancel out to result in 0.

Core Concepts Used

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Trigonometric Identities Algebraic Simplification

💡 EXAM TIP

Trigonometric simplification is frequently tested in competitive exams; always look for standard identity patterns like $1+tan^2\theta$ or $cosec^2\theta-1$ first.

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