Question

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

• A. $\frac{1}{2\sqrt{2}}$
• B. $\frac{1}{2}$
• C. $\frac{1}{4}$
• D. $\frac{1}{\sqrt{2}}$

Answer 1

The correct option is A

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

Explanation for correct option

We know that

$$\begin{gathered} \cos \left(3x\right)=4{\cos }^3\left(x\right)-3\cos \left(x\right) \\ \Rightarrow {\cos }^3\left(x\right)=\frac{1}{4}\left(\cos \left(3x\right)+3\cos \left(x\right)\right) \\ \sin \left(3x\right)=3\sin \left(x\right)-4{\sin }^3\left(x\right) \\ \Rightarrow {\sin }^3\left(x\right)=\frac{1}{4}\left(3\sin \left(x\right)-\sin \left(3x\right)\right) \end{gathered}$$

Using the above formula we get

$$\begin{gathered} {\cos }^3\left(\frac{\pi }{8}\right)·\cos \left(\frac{3\pi }{8}\right)+{\sin }^3\left(\frac{\pi }{8}\right)\sin \left(\frac{3\pi }{8}\right) \\ =\frac{1}{4}\left(\cos \left(\frac{3\pi }{8}\right)+3\cos \left(\frac{\pi }{8}\right)\right)\cos \left(\frac{3\pi }{8}\right)+\frac{1}{4}\left(3\sin \left(\frac{\pi }{8}\right)-\sin \left(\frac{3\pi }{8}\right)\right)\sin \left(\frac{3\pi }{8}\right) \\ =\frac{1}{4}\left({\cos }^2\left(\frac{3\pi }{8}\right)+3\cos \left(\frac{\pi }{8}\right)\cos \left(\frac{3\pi }{8}\right)+3\sin \left(\frac{\pi }{8}\right)\sin \left(\frac{3\pi }{8}\right)-{\sin }^2\left(\frac{3\pi }{8}\right)\right) \\ =\frac{1}{4}\left({\cos }^2\left(\frac{3\pi }{8}\right)-{\sin }^2\left(\frac{3\pi }{8}\right)\right)+\frac{3}{4}\left(\cos \left(\frac{\pi }{8}\right)\cos \left(\frac{3\pi }{8}\right)+\sin \left(\frac{\pi }{8}\right)\sin \left(\frac{3\pi }{8}\right)\right) \\ =\frac{1}{4}\left(\cos \left(\frac{3\pi }{4}\right)\right)+\frac{3}{4}\left(\cos \left(\frac{3\pi }{8}-\frac{\pi }{8}\right)\right) \\ =\frac{1}{4}\left(\cos \left(\frac{3\pi }{4}\right)\right)+\frac{3}{4}\left(\cos \left(\frac{\pi }{4}\right)\right) \\ =\frac{1}{4}\left(-\frac{1}{\sqrt{2}}\right)+\frac{3}{4\sqrt{2}} \\ =\frac{1}{2\sqrt{2}} \end{gathered}$$

Hence, the correct option is $\mathrm{Option}\left(\mathrm{A}\right)$