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NDA 2024
Exam
NDA
Year
2024
Subject
Mathematics
Chapter
Trigonometric Ratios & Identities

NDA 2024 · Mathematics · Trigonometric Ratios & Identities

If f(θ)=(1)/(1+ θ) and α+β=(5 π)/(4) , then what is the value of...

If f(θ)=11+tan⁡θf(\theta)=\frac{1}{1+\tan \theta} and α+β=5π4\alpha+\beta=\frac{5 \pi}{4}, then what is the value of f(α)f(β)f(\alpha) f(\beta) ?

Options

  1. A−12-\frac{1}{2}
  2. B12\frac{1}{2}
  3. C1
  4. D2

Correct answer

B. 12\frac{1}{2}

Step-by-step solution

α+β=5π4\alpha+\beta=\frac{5 \pi}{4}tan⁡(α+β)=tan⁡5π4⇒tan⁡α+tan⁡β1−tan⁡α×tan⁡β=tan⁡(π+π4)=tan⁡π4⇒tan⁡α+tan⁡β1−tan⁡α×tan⁡β=1⇒tan⁡α+tan⁡β=1−tan⁡α×tan⁡β⇒tan⁡α+tan⁡β+tan⁡α⋅tan⁡β=1∵f(θ)=11+tan⁡θf(α)⋅f(β)=11+tan⁡α+tan⁡β+tan⁡α⋅tan⁡β=11+1=12.\begin{aligned} & \tan (\alpha+\beta)=\tan 5 \frac{\pi}{4} \\ & \Rightarrow \frac{\tan \alpha+\tan \beta}{1-\tan \alpha \times \tan \beta}=\tan \left(\pi+\frac{\pi}{4}\right)=\tan \frac{\pi}{4} \\ & \Rightarrow \frac{\tan \alpha+\tan \beta}{1-\tan \alpha \times \tan \beta}=1 \\ & \Rightarrow \tan \alpha+\tan \beta=1-\tan \alpha \times \tan \beta \\ & \Rightarrow \tan \alpha+\tan \beta+\tan \alpha \cdot \tan \beta=1 \\ & \because f(\theta)=\frac{1}{1+\tan \theta} \\ & f(\alpha) \cdot f(\beta)=\frac{1}{1+\tan \alpha+\tan \beta+\tan \alpha \cdot \tan \beta}=\frac{1}{1+1}=\frac{1}{2}.\end{aligned}$$
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