Identify the correct expression from the given options. | Mathematics MCQ | Examoogle

Examoogle • User • info@examoogle.com • EE-2024-8821

Chapter 1 of 12 • Page 1 of 248🔒 Protected PDF • Watermarked

Back to Practice Questions

MathematicsTrigonometry

Prev Next

Identify the correct expression from the given options.

$0$

$1$

$2 \sec^2 A$

$\cot^2 A$

Correct Answer

📐 MA • Math Direct FormulaMathematicsTrigonometry

Option C

$2 \sec^2 A$

Quick Summary:

Observe the terms: $(\csc^2 A - 1) = \cot^2 A$ and $(1 + \tan^2 A) = \sec^2 A$ . The expression simplifies to $\sec^2 A + \cot^2 A + \sec^2 A - \cot^2 A$ , which reduces to $2\sec^2 A$ .

🎯

Crack Trigonometry with Examoogle SuperTeachers

Get Free Demo Classes & Live Mocks →

📐MAMath SolutionDirect Formula

📋 Given

An expression containing trigonometric identities: \sec² A + (\csc² A - 1) + (1 + $\tan$ ² A) - \cot² A

🔢 Formula Used

1+tan⁡2A=sec⁡2A1+cot⁡2A=csc⁡2A1 + \tan^2 A = \sec^2 A1 + \cot^2 A = \csc^2 A1+tan2A=sec2A1+cot2A=csc2A

⚡ Exam Hall Shortcut / Speed Trick

Observe the terms: $(\csc^2 A - 1) = \cot^2 A$ and $(1 + \tan^2 A) = \sec^2 A$ . The expression simplifies to $\sec^2 A + \cot^2 A + \sec^2 A - \cot^2 A$ , which reduces to $2\sec^2 A$ .

⚠️ Common Student Trap / Pitfall

Students often misremember the identity $\csc^2 A - 1 = \cot^2 A$ as $\tan^2 A$ or confuse the signs when expanding the terms.

🔢 Step-by-Step Solution

Simplify using identities

We know that $\csc^2 A - 1 = \cot^2 A$ and $1 + \tan^2 A = \sec^2 A$ . Substitute these identities into the expression.

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

Combine like terms

Group the terms $\sec^2 A$ and $\cot^2 A$ together. Notice that the positive $\cot^2 A$ and negative $\cot^2 A$ cancel out.

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

Final simplification

Adding the remaining terms gives the final result of $2\sec^2 A$ .

2sec⁡2A2\sec^2 A2sec2A

✅

C is correct because the trigonometric expression simplifies exactly to 2sec² A.

Core Concepts Used

Click any tag to open in AI Tutor

Trigonometric Identities Pythagorean Identities Simplification of Algebraic Expressions

💡 EXAM TIP

This concept of identity substitution is the foundation for solving complex trigonometric proofs in Calculus and Coordinate Geometry.

Was this solution helpful?

⚡ Practice on Android App · 4.8 ★

Solve 10,000+ More Questions with AI Tutor

Get instant step-by-step AI explanations, full shift Previous Year Papers (SSC GD, RRB, SSC JE), and regional rank analysis on the official Examoogle App.

Free Download • 100K+ Aspirants

Discussion (0)

Loading discussion...

Prev Next

Correct Answer

Solve 10,000+ More Questions with AI Tutor

Related Questions

Discussion (0)