# The value of cos π 65 cos 2 π 65 cos 4 π 65 cos 8 π 65 cos 16 π 65 cos 32 π 65 is - Mathematics

MCQ

The value of $\cos \frac{\pi}{65} \cos \frac{2\pi}{65} \cos \frac{4\pi}{65} \cos \frac{8\pi}{65} \cos \frac{16\pi}{65} \cos \frac{32\pi}{65}$  is

#### Options

• $\frac{1}{8}$

• $\frac{1}{16}$

• $\frac{1}{32}$

•  none of these

#### Solution

none of these

$\text{ We have } ,$
$\cos\frac{\pi}{65} \cos\frac{2\pi}{65} \cos\frac{4\pi}{65} \cos\frac{8\pi}{65} \cos\frac{16\pi}{65} \cos\frac{32\pi}{65}$
$= \frac{2\sin\frac{\pi}{65}}{2\sin\frac{\pi}{65}} \cos\frac{\pi}{65} \cos\frac{2\pi}{65} \cos\frac{4\pi}{65} \cos\frac{8\pi}{65} \cos\frac{16\pi}{65} \cos\frac{32\pi}{65}$
$\left( \text{ dividing and multiplying by } 2\sin\frac{\pi}{65} \right)$
$= \frac{2\sin\frac{2\pi}{65}}{2 \times 2\sin\frac{\pi}{65}} \cos\frac{2\pi}{65} \cos\frac{4\pi}{65} \cos\frac{8\pi}{65} \cos\frac{16\pi}{65} \cos\frac{32\pi}{65}$
$= \frac{2\sin\frac{4\pi}{65}}{2 \times 4\sin\frac{\pi}{65}} \cos\frac{4\pi}{65} \cos\frac{8\pi}{65} \cos\frac{16\pi}{65} \cos\frac{32\pi}{65}$

$= \frac{2\sin\frac{8\pi}{65}}{2 \times 8\sin\frac{\pi}{65}} \cos\frac{8\pi}{65} \cos\frac{16\pi}{65} \cos\frac{32\pi}{65}$
$= \frac{2\sin\frac{16\pi}{65}}{2 \times 16\sin\frac{\pi}{65}} \cos\frac{16\pi}{65} \cos\frac{32\pi}{65}$
$= \frac{2\sin\frac{32\pi}{65}}{2 \times 32\sin\frac{\pi}{65}} \cos\frac{32\pi}{65}$
$= \frac{\sin\frac{64\pi}{65}}{64\sin\frac{\pi}{65}}$
$= \frac{\sin\left( \pi - \frac{\pi}{65} \right)}{64\sin\frac{\pi}{65}}$
$= \frac{\sin\frac{\pi}{65}}{64\sin\frac{\pi}{65}}$
$= \frac{1}{64}$

Concept: Values of Trigonometric Functions at Multiples and Submultiples of an Angle
Is there an error in this question or solution?

#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 9 Values of Trigonometric function at multiples and submultiples of an angle
Q 3 | Page 43
Share