Value of \( \sin^2\frac{\pi}{18}+\sin^2\frac{\pi}{9}+\sin^2\frac{7\pi}{18}+\sin^2\frac{4\pi}{9} \)
Question
Find the value of
\[ \sin^2\frac{\pi}{18} +\sin^2\frac{\pi}{9} +\sin^2\frac{7\pi}{18} +\sin^2\frac{4\pi}{9} \]
(a) \(1\)
(b) \(2\)
(c) \(4\)
(d) none of these
Solution
Notice that
\[ \frac{7\pi}{18} = \frac{\pi}{2}-\frac{\pi}{9} \]
and
\[ \frac{4\pi}{9} = \frac{\pi}{2}-\frac{\pi}{18} \]
Using
\[ \sin\left(\frac{\pi}{2}-\theta\right)=\cos\theta \]
we get
\[ \sin^2\frac{7\pi}{18} = \cos^2\frac{\pi}{9} \]
\[ \sin^2\frac{4\pi}{9} = \cos^2\frac{\pi}{18} \]
Therefore,
\[ \sin^2\frac{\pi}{18} +\sin^2\frac{\pi}{9} +\cos^2\frac{\pi}{9} +\cos^2\frac{\pi}{18} \]
Grouping the terms,
\[ = \left( \sin^2\frac{\pi}{18} +\cos^2\frac{\pi}{18} \right) + \left( \sin^2\frac{\pi}{9} +\cos^2\frac{\pi}{9} \right) \]
Using
\[ \sin^2\theta+\cos^2\theta=1 \]
\[ =1+1 \]
\[ =2 \]
Final Answer
\[ \boxed{2} \]
Hence, the correct option is (b) 2.