# 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}$$+\cos^4\large\frac{7\pi}{8} \begin{array}{1 1}(A)\;\large\frac{1}{2}&(B)\;\large\frac{3}{2}\\(C)\;\large\frac{\sqrt 2}{2}&(D)\;\large\frac{1}{4}\end{array} ## 1 Answer Toolbox: • \cos(\pi-\theta)=\cos\theta • \cos(\large\frac{\pi}{2}$$-\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})]$