| The value of \[{{\sin }^{2}}1{}^\circ +{{\sin }^{2}}3{}^\circ +{{\sin }^{2}}5{}^\circ +...+{{\sin }^{2}}89{}^\circ \] is [SSC (10+2) 2013] |
A) \[22\]
B) \[22\frac{1}{2}\]
C) \[23\]
D) \[22\frac{1}{4}\]
Correct Answer: B
Solution :
| \[{{\sin }^{2}}1{}^\circ +{{\sin }^{2}}3{}^\circ +...+{{\sin }^{2}}45{}^\circ +...+\]\[{{\sin }^{2}}(90{}^\circ -3{}^\circ )+si{{n}^{2}}(90{}^\circ -1{}^\circ )\] |
| \[={{\sin }^{2}}1{}^\circ +{{\sin }^{2}}3{}^\circ +...+{{\sin }^{2}}45{}^\circ +...+\]\[{{\cos }^{2}}3{}^\circ +{{\cos }^{2}}1{}^\circ \] |
| \[=(1\times 22)+{{\left( \frac{1}{\sqrt{2}} \right)}^{2}}=22+\frac{1}{2}=22\frac{1}{2}\] |
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