4 cos frac{pi}{7} cos frac{pi}{5} cos frac{2 | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) We know the identity $\cos 	heta \cos 2	heta \cos 4	heta = \frac{\sin 8	heta}{8 \sin 	heta}$.First,consider the product $P_1 = \cos \frac{\pi

Vedclass · Accepted Answer

$-\frac{1}{8}$

Vedclass · Answer

(A) We know the identity $\cos 	heta \cos 2	heta \cos 4	heta = \frac{\sin 8	heta}{8 \sin 	heta}$.First,consider the product $P_1 = \cos \frac{\pi}{7} \cos \frac{2 \pi}{7} \cos \frac{4 \pi}{7}$.Using the identity with $	heta = \frac{\pi}{7}$,we get $P_1 = \frac{\sin(8\pi/7)}{8 \sin(\pi/7)} = \frac{\sin(\pi + \pi/7)}{8 \sin(\pi/7)} = \frac{-\sin(\pi/7)}{8 \sin(\pi/7)} = -\frac{1}{8}$.Next,consider the product $P_2 = \cos \frac{\pi}{5} \cos \frac{2 \pi}{5}$.Using the identity $\cos 	heta \cos 2	heta = \frac{\sin 4	heta}{4 \sin 	heta}$,we get $P_2 = \frac{\sin(4\pi/5)}{4 \sin(\pi/5)} = \frac{\sin(\pi - \pi/5)}{4 \sin(\pi/5)} = \frac{\sin(\pi/5)}{4 \sin(\pi/5)} = \frac{1}{4}$.Therefore,the total expression is $4 	imes P_1 	imes P_2 = 4 	imes (-\frac{1}{8}) 	imes (\frac{1}{4}) = -\frac{1}{8}$.

$4 \cos \frac{\pi}{7} \cos \frac{\pi}{5} \cos \frac{2 \pi}{7} \cos \frac{2 \pi}{5} \cos \frac{4 \pi}{7} = $

Similar Questions

If $A$ and $B$ are acute angles satisfying $3 \cos ^2 A + 2 \cos ^2 B = 4$ and $\frac{3 \sin A}{\sin B} = \frac{2 \cos B}{\cos A}$,then $A + 2B =$ (in $^{\circ}$)

Let $P = \{ \theta : \sin \theta - \cos \theta = \sqrt{2} \cos \theta \}$ and $Q = \{ \theta : \sin \theta + \cos \theta = \sqrt{2} \sin \theta \}$ be two sets. Then

$\cos^2 76^{\circ} + \cos^2 16^{\circ} - \cos 76^{\circ} \cos 16^{\circ}$ is equal to

$\text{If } \sin(\alpha+\beta)=1, \sin(\alpha-\beta)=\frac{1}{2}, \alpha, \beta \in [0, \frac{\pi}{2}], \text{ then } \tan(\alpha+2\beta) \cdot \tan(2\alpha+\beta) = ?$

Number of real values of $x \in (0, \pi)$ for which $\frac{8}{3\sin x - \sin 3x} + 3\sin^2 x \le 5$ is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module