Question

${\sin }^2\left(\frac{\pi }{8}\right)+{\sin }^2\left(\frac{3\pi }{8}\right)+{\sin }^2\left(\frac{5\pi }{8}\right)+{\sin }^2\left(\frac{7\pi }{8}\right)=$

• A. $1$
• B. $-1$
• C. $0$
• D. $2$

Answer 1

The correct option is D

$2$

Explanation for the correct answer:

Simplifying the given expression:

$$\begin{gathered} {\sin }^2\left(\frac{\pi }{8}\right)+{\sin }^2\left(\frac{3\pi }{8}\right)+{\sin }^2\left(\frac{5\pi }{8}\right)+{\sin }^2\left(\frac{7\pi }{8}\right) \\ ={\sin }^2\left(\frac{\pi }{8}\right)+{\sin }^2\left(\frac{3\pi }{8}\right)+{\sin }^2\left(\frac{\pi }{2}+\frac{\pi }{8}\right)+{\sin }^2\left(\frac{\pi }{2}+\frac{3\pi }{8}\right)\left[\because \sin \left(\frac{\pi }{2}+\theta \right)=\cos \theta \right] \\ ={\sin }^2\left(\frac{\pi }{8}\right)+{\sin }^2\left(\frac{3\pi }{8}\right)+{\cos }^2\left(\frac{\pi }{8}\right)+{\cos }^2\left(\frac{3\pi }{8}\right) \\ =\left[{\sin }^2\left(\frac{\pi }{8}\right)+{\cos }^2\left(\frac{\pi }{8}\right)\right]+\left[{\sin }^2\left(\frac{3\pi }{8}\right)+{\cos }^2\left(\frac{3\pi }{8}\right)\right] \\ =1+1\left[\because {\sin }^{2}x+{\cos }^{2}x=1\right] \\ =2 \end{gathered}$$

Therefore, the correct answer is option (D).