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NDA 2023
Exam
NDA
Year
2023
Subject
Mathematics
Chapter
Trigonometric Ratios & Identities

NDA 2023 · Mathematics · Trigonometric Ratios & Identities

Let p= ((π)/(5)) ((2 π)/(5)) and q= ((4 π)/(5)) ((8 π)/(5)). What...

Let p=cos⁡(π5)cos⁡(2π5)p=\cos \left(\frac{\pi}{5}\right) \cos \left(\frac{2 \pi}{5}\right)and q=\cos \left(\frac{4 \pi}{5}\right) \cos\left(\frac{8 \pi}{5}\right).WhatisthevalueofWhat is the value ofp q$ ?

Options

  1. A−116-\frac{1}{16}
  2. B−14-\frac{1}{4}
  3. C14\frac{1}{4}
  4. D116\frac{1}{16}

Correct answer

A. −116-\frac{1}{16}

Step-by-step solution

Since, pq =−cos⁡2(π5)⋅cos⁡2(2π5)=−(5+14)2⋅(5−14)2=−(5−1)216×16=−116\begin{aligned} & =-\cos ^2\left(\frac{\pi}{5}\right) \cdot \cos ^2\left(\frac{2 \pi}{5}\right) \\ & =-\left(\frac{\sqrt{5}+1}{4}\right)^2 \cdot\left(\frac{\sqrt{5}-1}{4}\right)^2 \\ & =-\frac{(5-1)^2}{16 \times 16}=-\frac{1}{16} \end{aligned}
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