Find the value of `sin^2(10) + sin^2(20) + sin^2(30) + … + sin^2(80)`.
\[ \text { Find the value of } \sin ^2 10+\sin ^2 20+\sin ^2 30+\ldots \ldots+\sin ^2 80 . \]
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Find the value of `sin^2(10) + sin^2(20) + sin^2(30) + … + sin^2(80)`.
Given:
– We need to find the value of \(\sin^2 10 + \sin^2 20 + \sin^2 30 + \ldots + \sin^2 80\).
1. We can pair the terms such that the sum of angles in each pair is \(90^\circ\):
\(\sin^2 10 + \sin^2 80, \sin^2 20 + \sin^2 70, \sin^2 30 + \sin^2 60, \sin^2 40 + \sin^2 50\)
2. Using the identity \(\sin^2 x + \sin^2 (90 – x) = 1\):
\(\sin^2 10 + \sin^2 80 = 1\)
\(\sin^2 20 + \sin^2 70 = 1\)
\(\sin^2 30 + \sin^2 60 = 1\)
\(\sin^2 40 + \sin^2 50 = 1\)
3. Adding these equations:
\(\sin^2 10 + \sin^2 20 + \sin^2 30 + \sin^2 40 + \sin^2 50 + \sin^2 60 + \sin^2 70 + \sin^2 80 = 4\)
Conclusion:
The value of \(\sin^2 10 + \sin^2 20 + \sin^2 30 + \ldots + \sin^2 80\) is 4.