Evaluate: left(1+cos frac{pi}{8}right)left(1+ | vedclass

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

Question

(C) Let the expression be $E = \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 \fr

Vedclass · Accepted Answer

$\frac{1}{8}$

Vedclass · Answer

(C) Let the expression be $E = \left(1+\cos \frac{\pi}{8}ight)\left(1+\cos \frac{3 \pi}{8}ight)\left(1+\cos \frac{5 \pi}{8}ight)\left(1+\cos \frac{7 \pi}{8}ight)$.Using the identity $\cos(\pi - 	heta) = -\cos 	heta$,we have $\cos \frac{5\pi}{8} = -\cos \frac{3\pi}{8}$ and $\cos \frac{7\pi}{8} = -\cos \frac{\pi}{8}$.Substituting these values,we get:$E = \left(1+\cos \frac{\pi}{8}ight)\left(1+\cos \frac{3 \pi}{8}ight)\left(1-\cos \frac{3 \pi}{8}ight)\left(1-\cos \frac{\pi}{8}ight)$$E = \left(1-\cos^2 \frac{\pi}{8}ight)\left(1-\cos^2 \frac{3 \pi}{8}ight)$$E = \sin^2 \frac{\pi}{8} \sin^2 \frac{3 \pi}{8}$Since $\frac{3\pi}{8} = \frac{\pi}{2} - \frac{\pi}{8}$,we have $\sin \frac{3\pi}{8} = \cos \frac{\pi}{8}$.$E = \sin^2 \frac{\pi}{8} \cos^2 \frac{\pi}{8} = \left(\sin \frac{\pi}{8} \cos \frac{\pi}{8}ight)^2$Using $2 \sin 	heta \cos 	heta = \sin 2	heta$,we get $\sin \frac{\pi}{8} \cos \frac{\pi}{8} = \frac{1}{2} \sin \frac{\pi}{4} = \frac{1}{2} 	imes \frac{1}{\sqrt{2}} = \frac{1}{2\sqrt{2}}$.$E = \left(\frac{1}{2\sqrt{2}}ight)^2 = \frac{1}{8}$.

Evaluate: $\left(1+\cos \frac{\pi}{8}\right)\left(1+\cos \frac{3 \pi}{8}\right)\left(1+\cos \frac{5 \pi}{8}\right)\left(1+\cos \frac{7 \pi}{8}\right)$

Similar Questions

The exact value of $\cos^2 73^\circ + \cos^2 47^\circ + (\cos 73^\circ \cdot \cos 47^\circ)$ is

If the equation $\tan^4x - 2\sec^2x + [a]^2 = 0$ has at least one solution,then the complete range of $a$ (where $a \in R$) is:
(Note: $[k]$ denotes the greatest integer less than or equal to $k$)

If $\cosh 2x = 199$,then $\operatorname{coth} x =$

If $\sin \theta \cosh \alpha = \tan x$ and $\cos \theta \sinh \alpha = \sec x$,then find the value of $\cos 2 \theta \cosh 2 \alpha$.

If $\tan \beta = \cos \theta \tan \alpha ,$ then ${\tan ^2}\frac{\theta }{2} = $

Vedclass Test Series

Exam Paper Generator

Online Exam Module