$(\cos 252^{0} - \sin 126^{0}) (\cos 252^{0} + \sin 126^{0}) (\sin^{2} 126^{0} + \sin^{2} 186 ° + \sin^{2} 66 °) =$

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$\frac{3 \sqrt{5}}{8}$

$- \frac{3 \sqrt{5}}{4}$

$\frac{3 \sqrt{5}}{4}$

$- \frac{3 \sqrt{5}}{8}$

answer is B.

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Detailed Solution

$\begin{array}{rcl} (\cos 252^{0} - \sin 126^{0}) (\cos 252^{0} + \sin 126^{0}) (\sin^{2} 126^{0}) + \sin^{2} 186 ° + \sin^{2} 66 °) \\ \begin{array}{cl} = & (\cos 252^{0} - \sin 126^{0}) (1 - \cos^{2} 186^{°} + \sin^{2} 126 ° + \sin^{2} 66 °) \\ = & \cos 378^{°} \cos 126^{°} [1 - (\cos^{2} 186^{°} - \sin^{2} 126^{°} + \sin^{2} 66^{°})] \\ = & \cos 18^{°} \cos (180^{°} - 54^{°}) [1 - \cos 312^{°} \cos 60^{°} + \frac{1 - \cos 132^{°}}{2}] \\ = & - \cos 18^{°} \cos 54^{°} (1 - \frac{1}{2} \cos 48^{°} + \frac{1}{2} + \frac{1}{2} \cos 48^{°}) \\ = & - \frac{\sqrt{10 + 2 \sqrt{5}}}{4} \frac{\sqrt{10 - 2 \sqrt{5}}}{4} (1 + \frac{1}{2}) \\ = & \frac{- \sqrt{100 - 20}}{16} (\frac{3}{2}) \end{array} \\ \begin{array}{cl} = & \frac{- \sqrt{80}}{16} \times \frac{3}{2} = \frac{(- 4 \sqrt{5}) 3}{32} \\ = & \frac{- 3 \sqrt{5}}{8} \end{array} \end{array}$