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

Back to Practice Questions

MathematicsPrevious Year Paper 2024

Prev Next

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

$2 \sec \theta$

$2 \cos \theta$

$2 \sin \theta$

$1 + 2 \sin \theta$

Correct Answer

📐 MA • Math Algebraic SimplificationMathematicsPrevious Year Paper 2024

Option A

$2 \sec \theta$

Quick Summary:

Plug in a convenient angle like $\theta = 0°$ . The expression becomes $\frac{1}{1} + \frac{1}{1} = 2$ . Checking the options at $\theta = 0°$ : A gives $2\sec(0)=2$ , B gives $2\cos(0)=2$ , C gives $2\sin(0)=0$ , D gives $1+0=1$ . Eliminating via $\theta=60°$ confirms A.

🎯

Crack Previous Year Paper 2024 with Examoogle SuperTeachers

Get Free Demo Classes & Live Mocks →

📐MAMath SolutionAlgebraic Simplification

📋 Given

An expression containing trigonometric functions: $\frac{\cos \theta}{1 - \sin \theta}$ $+ \frac{1 - \sin \theta}{\cos \theta}$ .

🔢 Formula Used

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

⚡ Exam Hall Shortcut / Speed Trick

⚠️ Common Student Trap / Pitfall

Students often struggle with the conjugate multiplication or common denominator approach, leading to sign errors when expanding the numerator $(1-\sin \theta)^2$ .

📊 Diagram / Illustration

🔢 Step-by-Step Solution

Find a common denominator

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

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​

Expand the numerator

Expand the squared term $(1 - \sin \theta)^2 = 1 - 2\sin \theta + \sin^2 \theta$ .

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​

Simplify using identities

Substitute $\sin^2 \theta + \cos^2 \theta = 1$ into the numerator: $1 + 1 - 2\sin \theta = 2(1 - \sin \theta)$ .

2(1−sin⁡θ)cos⁡θ(1−sin⁡θ)\frac{2(1 - \sin \theta)}{\cos \theta (1 - \sin \theta)}cosθ(1−sinθ)2(1−sinθ)​

Final reduction

Cancel the common term $(1 - \sin \theta)$ to get $\frac{2}{\cos \theta}$ , which is $2\sec \theta$ .

2cos⁡θ=2sec⁡θ\frac{2}{\cos \theta} = 2\sec \thetacosθ2​=2secθ

✅

A is correct because the algebraic simplification of the expression results in $2\sec \theta$ .

Core Concepts Used

Click any tag to open in AI Tutor

Trigonometric Identities Algebraic Fraction Addition Conjugate Method

💡 EXAM TIP

This type of simplification frequently appears in calculus where expressions are rewritten to make integration easier (e.g., rewriting trigonometric quotients as secant or tangent).

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)