Simplify the following expression:$\frac{\cos \theta}{1 + \si... | Mathematics MCQ | Examoogle

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Simplify the following expression: $\frac{\cos \theta}{1 + \sin \theta} + \frac{1 + \sin \theta}{\cos \theta}$

$2 \sec \theta$

$1 + 2 \sin \theta$

$2 \sin \theta$

$2 \cos \theta$

Correct Answer

📐 MA • Math Algebraic SimplificationMathematicsPrevious Year Paper 2024

Option A

$2 \sec \theta$

Quick Summary:

Test with a simple angle like $\theta = 0°$ . The expression becomes $\frac{1}{1} + \frac{1}{1} = 2$ . The options yield: A) $2\sec(0°) = 2$ , B) $1+0=1$ , C) $0$ , D) $2$ . Both A and D match, but simplify the algebraic form to identify the correct identity.

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📐MAMath SolutionAlgebraic Simplification

📋 Given

The expression is $\frac{\cos \theta}{1 + \sin \theta}$ $+ \frac{1 + \sin \theta}{\cos \theta}$

🔢 Formula Used

sin⁡2θ+cos⁡2θ=1,sec⁡θ=1cos⁡θ\sin^2 \theta + \cos^2 \theta = 1, \quad \sec \theta = \frac{1}{\cos \theta}sin2θ+cos2θ=1,secθ=cosθ1​

⚡ Exam Hall Shortcut / Speed Trick

⚠️ Common Student Trap / Pitfall

Students often expand the denominator $(1+\sin \theta)^2$ incorrectly or forget the fundamental Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$ .

📊 Diagram / Illustration

🔢 Step-by-Step Solution

Finding a common denominator

To add the fractions, find the common denominator $(1+\sin \theta)(\cos \theta)$ .

cos⁡2θ+(1+sin⁡θ)2cos⁡θ(1+sin⁡θ)\frac{\cos^2 \theta + (1+\sin \theta)^2}{\cos \theta (1+\sin \theta)}cosθ(1+sinθ)cos2θ+(1+sinθ)2​

Expanding the numerator

Expand $(1+\sin \theta)^2$ as $1 + 2\sin \theta + \sin^2 \theta$ .

cos⁡2θ+1+2sin⁡θ+sin⁡2θcos⁡θ(1+sin⁡θ)\frac{\cos^2 \theta + 1 + 2\sin \theta + \sin^2 \theta}{\cos \theta (1+\sin \theta)}cosθ(1+sinθ)cos2θ+1+2sinθ+sin2θ​

Applying trigonometric identity

Use the identity $\sin^2 \theta + \cos^2 \theta = 1$ to simplify the numerator.

(1)+1+2sin⁡θcos⁡θ(1+sin⁡θ)=2+2sin⁡θcos⁡θ(1+sin⁡θ)\frac{(1) + 1 + 2\sin \theta}{\cos \theta (1+\sin \theta)} = \frac{2 + 2\sin \theta}{\cos \theta (1+\sin \theta)}cosθ(1+sinθ)(1)+1+2sinθ​=cosθ(1+sinθ)2+2sinθ​

Final simplification

Factor out 2 from the numerator and cancel the $(1+\sin \theta)$ term.

2(1+sin⁡θ)cos⁡θ(1+sin⁡θ)=2cos⁡θ=2sec⁡θ\frac{2(1+\sin \theta)}{\cos \theta (1+\sin \theta)} = \frac{2}{\cos \theta} = 2\sec \thetacosθ(1+sinθ)2(1+sinθ)​=cosθ2​=2secθ

✅

A is correct because the algebraic simplification of the given expression leads to $2\sec \theta$ .

Core Concepts Used

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Trigonometric Identities Algebraic Fraction Addition Pythagorean Identity

💡 EXAM TIP

This concept of simplifying trigonometric expressions using common denominators is foundational for calculus, especially when dealing with trigonometric integration and derivatives.

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