If ${sin}^{2} 45 - {sin}^{2} 60 = x {cos}^{2} 45$ then $x =$ ..............

3/2

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3/4

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-1/2

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Solution

${sin}^{2} 45 - {sin}^{2} 60 = x {cos}^{2} 45$
$\Rightarrow {(\frac{1}{\sqrt{2}})}^{2} - {(\frac{\sqrt{3}}{2})}^{2} = x {(\frac{1}{\sqrt{2}})}^{2}$
$\Rightarrow {(\frac{\sqrt{2}}{2})}^{2} - {(\frac{\sqrt{3}}{2})}^{2} = x {(\frac{\sqrt{2}}{2})}^{2}$
$\Rightarrow \frac{2}{4} - \frac{3}{4} = \frac{2 x}{4}$
$\Rightarrow \frac{2 - 3}{4} = \frac{2 x}{4}$
$\Rightarrow - \frac{1}{4} = \frac{2 x}{4}$
$\Rightarrow - 1 = 2 x$
$\Rightarrow x = - \frac{1}{2}$
Hence, Option C is correct.

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