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NDA 2019
Exam
NDA
Year
2019
Subject
Mathematics
Chapter
Trigonometric Ratios & Identities

NDA 2019 · Mathematics · Trigonometric Ratios & Identities

If p=cosec θ- θ and q=(cosec θ+ θ)^ -1 , then which one of the...

If p=cosec⁡θ−cot⁡θp=\operatorname{cosec} \theta-\cot \theta and q=(cosec⁡θ+cot⁡θ)−1q=(\operatorname{cosec} \theta+\cot \theta)^{-1}, then which one of the following is correct?

Options

  1. Apq=1\mathrm{pq}=1
  2. Bp=q\mathrm{p}=\mathrm{q}
  3. Cp+q=1\mathrm{p}+\mathrm{q}=1
  4. Dp+q=0p+q=0

Correct answer

B. p=q\mathrm{p}=\mathrm{q}

Step-by-step solution

p=cosec⁡θ−cot⁡θ\mathrm{p}=\operatorname{cosec} \theta-\cot \thetaq=(cosec⁡θ+cot⁡θ)−1⇒1q=cosec⁡θ+cot⁡θ\begin{aligned} & \mathrm{q}=(\operatorname{cosec} \theta+\cot \theta)^{-1} \\ & \Rightarrow \frac{1}{q}=\operatorname{cosec} \theta+\cot \theta\end{aligned}We know, $\operatorname{cosec}^2 \theta-\cot ^2 \theta=1\begin{aligned} & \Rightarrow(\operatorname{cosec} \theta+\cot \theta)(\operatorname{cosec} \theta-\cot \theta)=1 \\ & \Rightarrow\left(\frac{1}{q}\right)(p)=1 \\ & \Rightarrow \mathrm{p}=\mathrm{q}\end{aligned}$$$
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