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If $\frac{\sin \theta + \cos \theta}{\sin \theta - \cos \theta} = 3$ , then the value of $\tan \theta$ is:

Correct Answer

📐 MA • Math Componendo and DividendoMathematicsTrigonometry

Option A

Quick Summary:

Use the property of Componendo and Dividendo directly: if (x+y)/(x-y) = k, then x/y = (k+1)/(k-1). Here, sin θ / cos θ = (3+1)/(3-1) = 4/2 = 2.

📐MAMath SolutionComponendo and Dividendo

📋 Given

The ratio of sum and difference of sin and cos is given as: (sin θ + cos θ) / (sin θ - cos θ) = 3

🔢 Formula Used

a+ba−b=cd  ⟹  ab=c+dc−d\frac{a+b}{a-b} = \frac{c}{d} \implies \frac{a}{b} = \frac{c+d}{c-d}a−ba+b​=dc​⟹ba​=c−dc+d​

⚡ Exam Hall Shortcut / Speed Trick

Use the property of Componendo and Dividendo directly: if (x+y)/(x-y) = k, then x/y = (k+1)/(k-1). Here, sin θ / cos θ = (3+1)/(3-1) = 4/2 = 2.

⚠️ Common Student Trap / Pitfall

Students often try to cross-multiply and solve for θ individually instead of identifying the tan θ ratio directly, leading to algebraic errors.

📊 Diagram / Illustration

🔢 Step-by-Step Solution

Apply Componendo and Dividendo

Given the equation $\frac{\sin \theta + \cos \theta}{\sin \theta - \cos \theta} = \frac{3}{1}$ , we apply the rule $\frac{a+b}{a-b} = \frac{c+d}{c-d}$ where $a = \sin \theta$ and $b = \cos \theta$ .

(sin⁡θ+cos⁡θ)+(sin⁡θ−cos⁡θ)(sin⁡θ+cos⁡θ)−(sin⁡θ−cos⁡θ)=3+13−1\frac{(\sin \theta + \cos \theta) + (\sin \theta - \cos \theta)}{(\sin \theta + \cos \theta) - (\sin \theta - \cos \theta)} = \frac{3+1}{3-1}(sinθ+cosθ)−(sinθ−cosθ)(sinθ+cosθ)+(sinθ−cosθ)​=3−13+1​

Simplify the fractions

After simplifying the numerator and denominator, the terms $\cos \theta$ and $\sin \theta$ cancel out respectively.

2sin⁡θ2cos⁡θ=42\frac{2 \sin \theta}{2 \cos \theta} = \frac{4}{2}2cosθ2sinθ​=24​

Final calculation

Since $\frac{\sin \theta}{\cos \theta} = \tan \theta$ , we divide the constants to reach the result.

tan⁡θ=2\tan \theta = 2tanθ=2

✅

A is correct because applying the Componendo and Dividendo rule leads directly to $\tan \theta = 2$ .

Core Concepts Used

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Trigonometric Ratios Componendo and Dividendo Definition of Tangent

💡 EXAM TIP

This property is frequently tested in 'Ratio and Proportion' chapters as well; identifying it saves time in competitive exams for problems involving linear combinations of functions.

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