What is the value of (1+ (π)/(8))(1+ (3 π)/(8))(1+ (5 π)/(8))(1+ (7 π)/(8)) ?

Correct answer: D. (1)/(8) [1+ (π)/(8)] [ 1+ (3 π)/(8)][1+ (5 π)/(8)] [ 1+ ( 7 π)/(8)] We have, aligned & (7 π)/(8)= [π-(π)/(8)]=- (π)/(8) & and (5 π)/(8)= [π-(3 π)/(8)]=- (3 π)/(8) & =[1+ (π)/(8)] [1+ (3 π)/(8)][1- (π)/(8)][1- (3 π)/(8)] & =[1- ^2 (π)/(8)][1- ^2 (3 π)/(8)]= ^2 (π)/(8) · ^2 (3 π)/(8) & =(1)/(4) [2 ^2 (π)/(8) ·2 ^2 (3 π)/(8)] & =(1)/(4)[ (1- (π)/(4))(1- (3 π)/(4))] & =(1)/(4)[ (1- 1 2 ) (1+ 1 2 )]=(1)/(8) aligned

NDA 2016

Exam: NDA
Year: 2016
Subject: Mathematics
Chapter: Trigonometric Ratios & Identities

NDA 2016 · Mathematics · Trigonometric Ratios & Identities

What is the value of (1+ (π)/(8))(1+ (3 π)/(8))(1+ (5 π)/(8))(1+ (7...

What is the value of \left(1+\cos \frac{\pi}{8}\right)\left(1+\cos \frac{3 \pi}{8}\right)\left(1+\cos \frac{5 \pi}{8}\right)\left(1+\cos \frac{7 \pi}{8}\right) ?

Options

A $\frac{1}{2}$
B $\frac{1}{2}+\frac{1}{2 \sqrt{2}}$
C $\frac{1}{2}-\frac{1}{2 \sqrt{2}}$
D $\frac{1}{8}$

Correct answer

D. $\frac{1}{8}$

Step-by-step solution

\left[1+\cos \frac{\pi}{8}\right] \left[

1+\cos

\frac{3 \pi}{8}\right]\left[1+\cos \frac{5 \pi}{8}\right] \left[

\frac{\cos 7 \pi}{8}\right] We have,

\begin{aligned}& \cos \frac{7 \pi}{8}=\cos \left[\pi-\frac{\pi}{8}\right]=-\cos \frac{\pi}{8}\\ & \text{\text{ and } } \cos \frac{5 \pi}{8}=\cos \left[\pi-\frac{3 \pi}{8}\right]=-\cos \frac{3 \pi}{8}\\ & \therefore=\left[1+\cos \frac{\pi}{8}\right] \left[1+\cos \frac{3 \pi}{8}\right]\left[1-\cos \frac{\pi}{8}\right]\left[1-\cos \frac{3 \pi}{8}\right]\\ & =\left[1-\cos ^2 \frac{\pi}{8}\right]\left[1-\cos ^2 \frac{3 \pi}{8}\right]=\sin ^2 \frac{\pi}{8} \cdot \sin ^2 \frac{3 \pi}{8}\\ & =\frac{1}{4} \left[2 \sin ^2 \frac{\pi}{8} \cdot2 \sin ^2 \frac{3 \pi}{8}\right]\\ & =\frac{1}{4}\left[ \left(1-\cos \frac{\pi}{4}\right)\left(1-\cos \frac{3 \pi}{4}\right)\right]\\ & =\frac{1}{4}\left[\lleft(1-\frac{1}{\sqrt{2}}\right)\lleft(1+\frac{1}{\sqrt{2}})]=\frac{1}{8} \end{aligned}

Practice Trigonometric Ratios & Identities on Quantrex Academy →

More from Trigonometric Ratios & Identities

If θ lies in the fourth quadrant and 3 θ+4=0 , then what is the value of 2θ+ 2θ ? NDA 2026 If α+ β=0= α+ β , α≠β then what is a value of 2α+ 2β+2 (α+β) ? NDA 2026 If 2 4β= 2α+ 2α , then which one of the following is a possible value of (α+β) ? NDA 2026 If x= θ- θ and y= θ+ θ , then which one of the following is correct ? NDA 2026 If θ=(1)/(3) , then what is the value of ((θ)/(2)) ((3θ)/(2)) ? NDA 2026 If A and B are acute angles such that 2A+2B=π , then what is the maximum value of A· B ? NDA 2026 x+ 3 x is maximum when x is equal to NDA 2026 In a triangle ABC , A= B+ C then what is ((B)/(2))+ ((B)/(2))equal to ? NDA 2026 If α and β are complementary angles such that α-β=(π)/(6) and m β=n α , then what is… NDA 2026 For the following two (02) items: Let A + B = p and A + B = q . What is (p^2 - q^2)/(p^2… NDA 2025 For the following two (02) items: Let p = cosec\, 20° and q = cosec\, 70° . What is (p^2… NDA 2025 For the following two (02) items: Let A + B = p and A + B = q . What is (p)/(q) equal to? NDA 2025 For the following two (02) items: Let p = cosec\, 20° and q = cosec\, 70° . What is ( 3… NDA 2025 Consider the following for the three (03) items that follow: Let p = 35° , q = 25° and r… NDA 2025 Consider the following for the three (03) items that follow: Let p = 35° , q = 25° and r… NDA 2025 Consider the following for the three (03) items that follow: Let p = 35° , q = 25° and r… NDA 2025 Full Trigonometric Ratios & Identities list Trigonometric Ratios & Identities JEE Maths PYQs JEE Mathematics PYQs All NDA PYQs