What is the value of ((5 π)/(17))+ ((7 π)/(17))+2 ((11 π)/(17)) ((π)/(17)) ?

Correct answer: A. 0 Concept: C+ D=2 ((C+D)/(2)) ((C-D)/(2)) Calculation: Given, ((5 π)/(17))+ ((7 π)/(17))+2 ((11 π)/(17)) ((π)/(17)) Using identity (i), aligned & 2 (((5 π)/(17)+(7 π)/(17))/(2)) (((5 π)/(17)-(7 π)/(17))/(2))+2 ((11 π)/(17)) ((π)/(17)) & 2 ((6 π)/(17)) (-(π)/(17))+2 ((11 π)/(17)) ((π)/(17)) & 2[ ((6 π)/(17)) ((π)/(17))+ ((11 π)/(17)) ((π)/(17))] & 2 ((π)/(17))[ ((6 π)/(17))+ ((11 π)/(17))] aligned Again using (i), aligned & 2 ((π)/(17))[2 (((6 π)/(17)+(11 π)/(17))/(2)) (((6 π)/(17)-(11 π)/(17))/(2))] & 4 ((π)/(17)) ((π)/(2)) ((5 π)/(34)) & 4 ((π)/(17)) ×0 × ((5 π)/(34))=0 aligned

NDA 2022

Exam: NDA
Year: 2022
Subject: Mathematics
Chapter: Trigonometric Ratios & Identities

NDA 2022 · Mathematics · Trigonometric Ratios & Identities

What is the value of ((5 π)/(17))+ ((7 π)/(17))+2 ((11 π)/(17))...

What is the value of

\cos \left(\frac{5 \pi}{17}\right)+\cos \left(\frac{7 \pi}{17}\right)+2 \cos \left(\frac{11 \pi}{17}\right) \cos \left(\frac{\pi}{17}\right) ?

Options

A $0$
B $1$
C $4 \cos \left(\frac{6 \pi}{17}\right) \cos \left(\frac{\pi}{17}\right)$
D $4 \cos \left(\frac{11 \pi}{17}\right) \cos \left(\frac{\pi}{17}\right)$

Correct answer

A. $0$

Step-by-step solution

Concept:

\cos C+\cos \mathrm{D}=2 \cos

\left(\frac{C+D}{2}\right) \cos

\left(\frac{C-D}{2}\right) Calculation: Given, \cos \left(\frac{5 \pi}{17}\right)+\cos \left(\frac{7 \pi}{17}\right)+2 \cos \left(\frac{11 \pi}{17}\right) \cos \left(\frac{\pi}{17}\right)

Using identity (i),

$\begin{aligned}& \Rightarrow 2 \cos \left(\frac{\frac{5 \pi}{17}+\frac{7 \pi}{17}}{2}\right) \cos \left(\frac{\frac{5 \pi}{17}-\frac{7 \pi}{17}}{2}\right)+2 \cos \left(\frac{11 \pi}{17}\right) \cos \left(\frac{\pi}{17}\right)\\ & \Rightarrow 2 \cos \left(\frac{6 \pi}{17}\right) \cos \left(-\frac{\pi}{17}\right)+2 \cos \left(\frac{11 \pi}{17}\right) \cos \left(\frac{\pi}{17}\right)\\ & \Rightarrow 2\left[\cos \left(\frac{6 \pi}{17}\right) \cos \left(\frac{\pi}{17}\right)+\cos \left(\frac{11 \pi}{17}\right) \cos \left(\frac{\pi}{17}\right)\right]\\ & \Rightarrow 2 \cos \left(\frac{\pi}{17}\right)\left[\cos \left(\frac{6 \pi}{17}\right)+\cos \left(\frac{11 \pi}{17}\right)\right] \end{aligned}

Again using (i),

\begin{aligned}& \Rightarrow 2 \cos \left(\frac{\pi}{17}\right)\left[2 \cos \left(\frac{\frac{6 \pi}{17}+\frac{11 \pi}{17}}{2}\right) \cos\left(\frac{\frac{6 \pi}{17}-\frac{11 \pi}{17}}{2}\right)\right]\\ & \Rightarrow 4 \cos \left(\frac{\pi}{17}\right) \cos \left(\frac{\pi}{2}\right) \cos \left(\frac{5 \pi}{34}\right)\\ & \Rightarrow 4 \cos \left(\frac{\pi}{17}\right) \times0 \times \cos \left(\frac{5 \pi}{34}\right)=0 \end{aligned}

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