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Home  >>  CBSE XI  >>  Math  >>  Trigonometric Functions
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Find the value of the expression :$\cos^4\large\frac{\pi}{8}$$+\cos^4\large\frac{3\pi}{8}$$+\cos^4\large\frac{5\pi}{8}$$+\cos^4\large\frac{7\pi}{8}$

$\begin{array}{1 1}(A)\;\large\frac{1}{2}&(B)\;\large\frac{3}{2}\\(C)\;\large\frac{\sqrt 2}{2}&(D)\;\large\frac{1}{4}\end{array} $

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1 Answer

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Toolbox:
  • $\cos(\pi-\theta)=\cos\theta$
  • $\cos(\large\frac{\pi}{2}$$-\theta)=\sin\theta$
  • $\sin^2\theta+\cos^2\theta=1$
  • $\sin^2\theta=2\sin \theta\cos\theta$
$\cos^4\large\frac{\pi}{8}$$+\cos^4\large\frac{3\pi}{8}$$+\cos^4\large\frac{5\pi}{8}$$+\cos^4\large\frac{7\pi}{8}$
$\Rightarrow \cos^4\large\frac{\pi}{8}$$+\cos^4\large\frac{3\pi}{8}$$+\cos^4(\pi-\large\frac{3\pi}{8})$$+\cos^4(\pi-\large\frac{\pi}{8})$
$\Rightarrow\cos^4\large\frac{\pi}{8}$$+\cos^4\large\frac{3\pi}{8}$$+\cos^4\large\frac{3\pi}{8}$$+\cos\large\frac{\pi}{8}$
$\Rightarrow2(\cos^4\large\frac{\pi}{8}+$$\cos^4\large\frac{3\pi}{8})$
$\Rightarrow 2[(\cos^2\large\frac{\pi}{8}$$+\cos^2\large\frac{3\pi}{8})^2-$$2\cos^2\large\frac{\pi}{8}$$\cos^2\large\frac{3\pi}{8}]$
$\Rightarrow [(\cos^2\large\frac{\pi}{8}$$+\cos^2(\large\frac{\pi}{2}-\frac{\pi}{8})$$-2\cos^2\large\frac{\pi}{8}$$\cos^2(\large\frac{\pi}{2}-\frac{\pi}{8})]$
$\Rightarrow 2[(\cos^2\large\frac{\pi}{8}$$+\sin^2\large\frac{\pi}{8})$$-2\cos^2\large\frac{\pi}{8}$$\sin^2\large\frac{\pi}{8}]$
$\Rightarrow 2[1-2\large\frac{\sin^2\Large\frac{\pi}{8}}{4}]$
$\Rightarrow 2[1-(\large\frac{\sqrt 2}{2})^2\times \large\frac{1}{2}]$
$\Rightarrow 2\times [1-\large\frac{1}{2}\times \large\frac{1}{2}]$
$\Rightarrow 2\times \large\frac{3}{4}=\large\frac{3}{2}$
Hence (B) is the correct answer.
answered Apr 21, 2014 by sreemathi.v
 

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