Simplify the following expression:$\frac{\sin \theta}{1 + \co... | Mathematics MCQ

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

$2 \cos \theta$

$2 \csc \theta$

$2 \sec \theta$

$2 \sin \theta$

Correct Answer

📐 MA • Math Direct FormulaMathematicsPrevious Year Paper 2024

Option B

$2 \csc \theta$

Quick Summary:

Use the identity $\sin^2 \theta + \cos^2 \theta = 1$ . When you cross-multiply, the numerator becomes $\sin^2 \theta + 1 + 2 \cos \theta + \cos^2 \theta = 2 + 2 \cos \theta = 2(1 + \cos \theta)$ . This cancels the $(1 + \cos \theta)$ in the denominator, leaving $2/\sin \theta = 2 \csc \theta$ .

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📐MAMath SolutionDirect Formula

📋 Given

Expression to simplify: (sin θ / (1 + cos θ)) + ((1 + cos θ) / sin θ)

🔢 Formula Used

sin⁡θ1+cos⁡θ+1+cos⁡θsin⁡θ=sin⁡2θ+(1+cos⁡θ)2sin⁡θ(1+cos⁡θ)\frac{\sin \theta}{1 + \cos \theta} + \frac{1 + \cos \theta}{\sin \theta} = \frac{\sin^2 \theta + (1 + \cos \theta)^2}{\sin \theta (1 + \cos \theta)}1+cosθsinθ​+sinθ1+cosθ​=sinθ(1+cosθ)sin2θ+(1+cosθ)2​

⚡ Exam Hall Shortcut / Speed Trick

⚠️ Common Student Trap / Pitfall

Students often forget to expand $(1 + \cos \theta)^2$ correctly as $1 + 2 \cos \theta + \cos^2 \theta$ , or they try to cancel terms across the addition before finding a common denominator.

🔢 Step-by-Step Solution

Find the common denominator

To add the fractions, find a common denominator by multiplying the two denominators: $\sin \theta (1 + \cos \theta)$ .

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

Expand the numerator

Expand the term $(1 + \cos \theta)^2$ as $1 + 2 \cos \theta + \cos^2 \theta$ and group $\sin^2 \theta + \cos^2 \theta$ .

sin⁡2θ+cos⁡2θ+1+2cos⁡θ=1+1+2cos⁡θ=2+2cos⁡θ\sin^2 \theta + \cos^2 \theta + 1 + 2 \cos \theta = 1 + 1 + 2 \cos \theta = 2 + 2 \cos \thetasin2θ+cos2θ+1+2cosθ=1+1+2cosθ=2+2cosθ

Simplify and factor

Factor out 2 from the numerator and cancel the common factor $(1 + \cos \theta)$ from numerator and denominator.

2(1+cos⁡θ)sin⁡θ(1+cos⁡θ)=2sin⁡θ\frac{2(1 + \cos \theta)}{\sin \theta (1 + \cos \theta)} = \frac{2}{\sin \theta}sinθ(1+cosθ)2(1+cosθ)​=sinθ2​

Final conversion

Using the trigonometric identity $1/\sin \theta = \csc \theta$ , we get the final result.

2csc⁡θ2 \csc \theta2cscθ

✅

B is correct because the expression simplifies to 2 cosec θ.

Core Concepts Used

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Trigonometric Identities Fractional Addition Simplification

💡 EXAM TIP

This problem demonstrates the importance of the Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$ , which is a fundamental tool used in calculus and complex number problems involving rotation.

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