• # question_answer The value of ${{\sin }^{2}}{{5}^{o}}+{{\sin }^{2}}{{10}^{o}}+{{\sin }^{2}}{{15}^{o}}+...+$ ${{\sin }^{2}}{{85}^{o}}+{{\sin }^{2}}{{90}^{o}}$ is  equal to [Karnataka CET 1999] A) 7 B) 8 C) 9 D) $9\frac{1}{2}$

Correct Answer: D

Solution :

Given expression is ${{\sin }^{2}}{{5}^{o}}+{{\sin }^{2}}{{10}^{o}}+{{\sin }^{2}}{{15}^{o}}+.....+{{\sin }^{2}}{{85}^{o}}+{{\sin }^{2}}{{90}^{o}}.$ We know that $\sin {{90}^{o}}=1$ or ${{\sin }^{2}}{{90}^{o}}=1$. Similarly, $\sin {{45}^{o}}=\frac{1}{\sqrt{2}}\text{or}\,\text{si}{{\text{n}}^{\text{2}}}{{45}^{o}}=\frac{1}{2}$ and the angles are in A.P. of 18 terms. We also know that ${{\sin }^{2}}{{85}^{o}}={{[\sin ({{90}^{o}}-{{5}^{o}})]}^{2}}$$={{\cos }^{2}}{{5}^{o}}.$ Therefore from the complementary rule, we find ${{\sin }^{2}}{{5}^{o}}+{{\sin }^{2}}{{85}^{o}}={{\sin }^{2}}{{5}^{o}}+{{\cos }^{2}}{{5}^{o}}=1.$ Therefore, ${{\sin }^{2}}{{5}^{o}}+{{\sin }^{2}}{{10}^{o}}+{{\sin }^{2}}{{15}^{o}}+...+{{\sin }^{2}}{{85}^{o}}+{{\sin }^{2}}{{90}^{o}}$ $=(1+1+1+1+1+1+1+1)+1+\frac{1}{2}=9\frac{1}{2}$.

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