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
Correct Option: , 2
$\cos ^{3} \frac{\pi}{8}\left[4 \cos ^{3} \frac{\pi}{8}-3 \cos \frac{\pi}{8}\right]$
$+\sin ^{3} \frac{\pi}{8}\left[3 \sin \frac{\pi}{8}-4 \sin ^{3} \frac{\pi}{8}\right]$
$=4 \cos ^{6} \frac{\pi}{8}-4 \sin ^{6} \frac{\pi}{8}-3 \cos ^{4} \frac{\pi}{8}+3 \sin ^{4} \frac{\pi}{8}$
$=4\left[\left(\cos ^{2} \frac{\pi}{8}-\sin ^{2} \frac{\pi}{8}\right)\right]$
$\left[\left(\sin ^{4} \frac{\pi}{8}+\cos ^{4} \frac{\pi}{8}+\sin ^{2} \frac{\pi}{8} \cos ^{2} \frac{\pi}{8}\right)\right]$
$-3\left[\left(\cos ^{2} \frac{\pi}{8}-\sin ^{2} \frac{\pi}{8}\right)\left(\cos ^{2} \frac{\pi}{8}+\sin ^{2} \frac{\pi}{8}\right)\right]$
$=\cos \frac{\pi}{4}\left[4\left(1-\sin ^{2} \frac{\pi}{8} \cos ^{2} \frac{\pi}{8}\right)-3\right]$
$=\frac{1}{\sqrt{2}}\left[1-\frac{1}{2}\right]=\frac{1}{2 \sqrt{2}}$
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