If $\dfrac{\sin \theta + \cos \theta}{\sin \theta - \cos \thet... | Mathematics MCQ | Examoogle

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If $\dfrac{\sin \theta + \cos \theta}{\sin \theta - \cos \theta} = 11$ , then the value of $\dfrac{25 \tan^2 \theta - 3}{36 \cot^2 \theta + 2} - 1$ is:

2/9

11/9

7/9

4/9

Correct Answer

📐 MA • Math Trigonometric SubstitutionMathematicsTrigonometry

Option B

11/9

Quick Summary:

Divide numerator and denominator by cosθ to get (tanθ+1)/(tanθ-1) = 11, then solve for tanθ. Substitute tanθ = 6/5 and cotθ = 5/6 into the target expression.

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📐MAMath SolutionTrigonometric Substitution

📋 Given

The ratio (sinθ + cosθ) / (sinθ - cosθ) = 11.

🔢 Formula Used

sin⁡θ+cos⁡θsin⁡θ−cos⁡θ=1+cot⁡θ1−cot⁡θ=11\frac{\sin \theta + \cos \theta}{\sin \theta - \cos \theta} = \frac{1 + \cot \theta}{1 - \cot \theta} = 11sinθ−cosθsinθ+cosθ​=1−cotθ1+cotθ​=11

⚡ Exam Hall Shortcut / Speed Trick

Divide numerator and denominator by cosθ to get (tanθ+1)/(tanθ-1) = 11, then solve for tanθ. Substitute tanθ = 6/5 and cotθ = 5/6 into the target expression.

⚠️ Common Student Trap / Pitfall

Many students fail to divide by cosθ and try to solve for sinθ and cosθ individually, which leads to unnecessarily long algebraic equations.

📊 Diagram / Illustration

🔢 Step-by-Step Solution

Convert to tanθ

Divide numerator and denominator of the given equation by $\cos \theta$ : $\frac{\tan \theta + 1}{\tan \theta - 1} = 11$

tan⁡θ+1=11tan⁡θ−11  ⟹  10tan⁡θ=12  ⟹  tan⁡θ=65\tan \theta + 1 = 11 \tan \theta - 11 \implies 10 \tan \theta = 12 \implies \tan \theta = \frac{6}{5}tanθ+1=11tanθ−11⟹10tanθ=12⟹tanθ=56​

Find cotθ

Since $\cot \theta = \frac{1}{\tan \theta}$ , we calculate the reciprocal: $\cot \theta = \frac{5}{6}$

cot⁡θ=56\cot \theta = \frac{5}{6}cotθ=65​

Evaluate the target expression

Substitute $\tan^2 \theta = (\frac{6}{5})^2 = \frac{36}{25}$ and $\cot^2 \theta = (\frac{5}{6})^2 = \frac{25}{36}$ into the expression $\frac{25 \tan^2 \theta - 3}{36 \cot^2 \theta + 2} - 1$

25(3625)−336(2536)+2−1=36−325+2−1=3327−1\frac{25(\frac{36}{25}) - 3}{36(\frac{25}{36}) + 2} - 1 = \frac{36 - 3}{25 + 2} - 1 = \frac{33}{27} - 136(3625​)+225(2536​)−3​−1=25+236−3​−1=2733​−1

Final Calculation

Simplify the fraction $\frac{33}{27} = \frac{11}{9}$ and subtract 1: $\frac{11}{9} - 1 = \frac{2}{9}$

11−99=29\frac{11 - 9}{9} = \frac{2}{9}911−9​=92​

✅

B is correct because the evaluated result of the expression is 2/9.

Core Concepts Used

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Trigonometric identities Algebraic simplification Reciprocal ratios

💡 EXAM TIP

Similar techniques involving dividing by $\cos \theta$ are frequently used in problems involving $\sin \theta$ and $\cos \theta$ sums/differences to convert them into $\tan \theta$ form.

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