If x sin 45
Question:

If $x \sin 45^{\circ} \cos ^{2} 60^{\circ}=\frac{\tan ^{2} 60^{\circ} \operatorname{cosec} 30^{\circ}}{\sec 45^{\circ} \cot ^{2} 30^{\circ}}$, then $x=$

(a) 2

(b) 4

(c) 8

(d) 16

Solution:

(c) 8

We have:

$x \sin 45^{\circ} \cos ^{2} 60^{\circ}=\frac{\tan ^{2} 60^{\circ} \operatorname{cosec} 30^{\circ}}{\sec 45^{\circ} \cot ^{2} 30^{\circ}}$

$\Rightarrow x \times\left(\frac{1}{\sqrt{2}}\right) \times\left(\frac{1}{2}\right)^{2}=\frac{(\sqrt{3})^{2} \times(2)}{(\sqrt{2}) \times(\sqrt{3})^{2}}$

$\Rightarrow \frac{x}{4 \sqrt{2}}=\frac{6}{3 \sqrt{2}}$

$\Rightarrow x=\frac{6}{3 \sqrt{2}} \times 4 \sqrt{2}$

$\Rightarrow x=8$

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