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The value of $\sin ^2 5^{\circ} + \sin ^2 10^{\circ} + \sin ^2 15^{\circ} + \dots + \sin^2 90^{\circ}$ is

  1. $8$
  2. $9$
  3. $9.5$
  4. None of these
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$\underline{\textbf{Answer:C}\Rightarrow}$

$\underline{\textbf{Solution:}\Rightarrow}$

We will make the pairs of $\sin^2\theta + \cos^2\theta$, where $\theta$ is ame for both $\sin$ and $\cos$.

$\sin^25^0 + \sin^285^0 = \sin^25^0 + \sin^2(90^0-5^0) = \sin^25^0 + \cos^25^0 =1\;\;\;\mathrm{[Using\;\;sin^2\theta + \cos^2\theta = 1]}$

$\sin^2{85}^0 = \cos^25^0 (\color{blue}{\because \sin\theta = \cos({90^0-\theta})})$

$\text{Similarly,}\;\;\;\sin^210 \;\text{and}\; \sin^280$

$\vdots$

$\sin^245^0 = \bigg(\frac{1}{\sqrt2}\bigg)^2 = \frac{1}{2} = 0.5$

$\vdots$

$\sin^240^0 \;\text{and}\; \sin^250$

$\vdots$

$\sin^290^0 = (1)^2 = 1$

So, this way $\mathbf 8$ pairs will be formed, leaving $\sin^245^0$ and $\sin^290^0$

$\text{But we know that:}\;\;\;\sin^245^0 = \bigg(\frac{1}{\sqrt2}\bigg)^2 = \frac{1}{2} = 0.5$

$\sin^290^0 = 1^2 = 1$

So, answer $= 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 +0.5 =  8\times1 +0.5 =9.5 $

$\therefore \mathbf C$ is the right option.
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Edit:- answer is 9.5, not 8.5
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Yes!

Thanks.
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