The value of cos^{3}left(frac{pi}{8}right) cd | vedclass

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Question

(C) Let $	heta = \frac{\pi}{8}$. Then $\frac{3\pi}{8} = 3	heta$. Given expression is $\cos^{3}(	heta) \cos(3	heta) + \sin^{3}(	heta) \sin(3	heta

Vedclass · Accepted Answer

$\frac{1}{2\sqrt{2}}$

Vedclass · Answer

(C) Let $	heta = \frac{\pi}{8}$. Then $\frac{3\pi}{8} = 3	heta$. Given expression is $\cos^{3}(	heta) \cos(3	heta) + \sin^{3}(	heta) \sin(3	heta)$. Using the identities $\cos(3	heta) = 4\cos^{3}(	heta) - 3\cos(	heta)$ and $\sin(3	heta) = 3\sin(	heta) - 4\sin^{3}(	heta)$,we can rewrite the expression. Alternatively,note that $\cos(\frac{3\pi}{8}) = \sin(\frac{\pi}{8})$ and $\sin(\frac{3\pi}{8}) = \cos(\frac{\pi}{8})$. Substituting these: $\cos^{3}(\frac{\pi}{8}) \sin(\frac{\pi}{8}) + \sin^{3}(\frac{\pi}{8}) \cos(\frac{\pi}{8})$. Factor out $\sin(\frac{\pi}{8}) \cos(\frac{\pi}{8})$: $= \sin(\frac{\pi}{8}) \cos(\frac{\pi}{8}) [\cos^{2}(\frac{\pi}{8}) + \sin^{2}(\frac{\pi}{8})]$. Since $\cos^{2}(	heta) + \sin^{2}(	heta) = 1$,this simplifies to $\sin(\frac{\pi}{8}) \cos(\frac{\pi}{8})$. Using $\sin(2	heta) = 2\sin(	heta)\cos(	heta)$,we get $\frac{1}{2} \sin(2 \cdot \frac{\pi}{8}) = \frac{1}{2} \sin(\frac{\pi}{4})$. $= \frac{1}{2} \cdot \frac{1}{\sqrt{2}} = \frac{1}{2\sqrt{2}}$.

The value of $\cos^{3}\left(\frac{\pi}{8}\right) \cdot \cos\left(\frac{3\pi}{8}\right) + \sin^{3}\left(\frac{\pi}{8}\right) \cdot \sin\left(\frac{3\pi}{8}\right)$ is:

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