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NDA 2021
Exam
NDA
Year
2021
Subject
Mathematics
Chapter
Trigonometric Ratios & Identities

NDA 2021 · Mathematics · Trigonometric Ratios & Identities

What is (1+ θ-cosec θ)(1+ θ+ θ) equal to?

What is (1+cot⁡θ−cosec⁡θ)(1+tan⁡θ+sec⁡θ)(1+\cot \theta-\operatorname{cosec} \theta)(1+\tan \theta+\sec \theta) equal to?

Options

  1. A11
  2. B22
  3. C33
  4. D44

Correct answer

B. 22

Step-by-step solution

(1+cot⁡θ−cosec⁡θ)(1+tan⁡θ+sec⁡θ)=(sin⁡θ+cos⁡θ−1sin⁡θ)(sin⁡θ+cos⁡θ+1cos⁡θ)\begin{aligned} & (1+\cot \theta-\operatorname{cosec} \theta)(1+\tan \theta+\sec \theta) \\ & =\left(\frac{\sin \theta+\cos \theta-1}{\sin \theta}\right)\left(\frac{\sin \theta+\cos \theta+1}{\cos \theta}\right)\end{aligned}=(sin⁡θ+cos⁡θ)2−12sin⁡θ⋅cos⁡θ=sin⁡2θ+cos⁡2θ+2sin⁡θ⋅cos⁡θ−1sin⁡θ⋅cos⁡θ\begin{aligned} & =\frac{(\sin \theta+\cos \theta)^2-1^2}{\sin \theta \cdot \cos \theta} \\ & =\frac{\sin ^2 \theta+\cos ^2 \theta+2 \sin \theta \cdot \cos \theta-1}{\sin \theta \cdot \cos \theta}\end{aligned}=\frac{1+2 \sin \theta \cdot \cos \theta-1}{\sin \theta \cdot \cos \theta}=2$
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