96 ( 33) ( 2 33) ( 4 33) ( 8 33) ( 16 33) is equal to: — Solution | JEE Mathematics

MCQEasyJEE 2023Trigonometric Equations

$96 \cos\left(\frac{\pi}{33}\right) \cos\left(\frac{2\pi}{33}\right) \cos\left(\frac{4\pi}{33}\right) \cos\left(\frac{8\pi}{33}\right) \cos\left(\frac{16\pi}{33}\right)$ is equal to:

A
$4$
B
$2$
C
$3$
D
$1$

Answer

Correct answer:C

Step-by-step solution

Standard Method

Given:

96 \cos\left(\frac{\pi}{33}\right) \cos\left(\frac{2\pi}{33}\right) \cos\left(\frac{4\pi}{33}\right) \cos\left(\frac{8\pi}{33}\right) \cos\left(\frac{16\pi}{33}\right)

Find: The value of the given expression.

Use the identity

\cos A \cos 2A \cos 2^2 A \dots \cos 2^{n-1} A = \frac{\sin(2^n A)}{2^n \sin A}

Here,

A = \frac{\pi}{33}, \qquad n = 5

So the expression becomes

96 \cdot \frac{\sin\left(2^5 \cdot \frac{\pi}{33}\right)}{2^5 \sin\left(\frac{\pi}{33}\right)}

Since

2^5 = 32

we get

96 \cdot \frac{\sin\left(\frac{32\pi}{33}\right)}{32 \sin\left(\frac{\pi}{33}\right)}

Now use

\sin(\pi - x) = \sin x

Therefore,

\sin\left(\frac{32\pi}{33}\right) = \sin\left(\frac{\pi}{33}\right)

Substituting,

96 \cdot \frac{\sin\left(\frac{\pi}{33}\right)}{32 \sin\left(\frac{\pi}{33}\right)} = \frac{96}{32} = 3

Therefore, the correct option is C.

Common mistakes

Using the product identity with the wrong value of $n$ . There are 5 cosine factors, so $n = 5$ . Counting incorrectly changes the power of $2$ in the denominator and gives a wrong result.
Not recognizing that $\sin\left(\frac{32\pi}{33}\right) = \sin\left(\frac{\pi}{33}\right)$ because $\frac{32\pi}{33} = \pi - \frac{\pi}{33}$ . Use the identity $\sin(\pi - x) = \sin x$ before simplifying.
Cancelling terms before rewriting the numerator correctly. First transform the product into a sine ratio, then apply the identity, and only after that cancel the common $\sin\left(\frac{\pi}{33}\right)$ factor.

Practice more Trigonometric Equations questions

Get unlimited AI-adaptive practice, mastery tracking, and an AI tutor that explains every step — free to start.

Start free trial More on Trigonometric Equations

Answer

Step-by-step solution

Common mistakes

Practice more Trigonometric Equations questions

Related questions