The numerical value of frac{cos ^{2} 45^{circ | vedclass

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Question

(A) Given expression: $\frac{\cos ^{2} 45^{\circ}}{\sin ^{2} 60^{\circ}}+\frac{\cos ^{2} 60^{\circ}}{\sin ^{2} 45^{\circ}}-\frac{	an ^{2} 30^{\circ}}

Vedclass · Accepted Answer

$\frac{3}{4}$

Vedclass · Answer

(A) Given expression: $\frac{\cos ^{2} 45^{\circ}}{\sin ^{2} 60^{\circ}}+\frac{\cos ^{2} 60^{\circ}}{\sin ^{2} 45^{\circ}}-\frac{	an ^{2} 30^{\circ}}{\cot ^{2} 45^{\circ}}-\frac{\sin ^{2} 30^{\circ}}{\cot ^{2} 30^{\circ}}$Substitute the trigonometric values: $\cos 45^{\circ} = \frac{1}{\sqrt{2}}$,$\sin 60^{\circ} = \frac{\sqrt{3}}{2}$,$\cos 60^{\circ} = \frac{1}{2}$,$\sin 45^{\circ} = \frac{1}{\sqrt{2}}$,$	an 30^{\circ} = \frac{1}{\sqrt{3}}$,$\cot 45^{\circ} = 1$,$\sin 30^{\circ} = \frac{1}{2}$,$\cot 30^{\circ} = \sqrt{3}$.$\Rightarrow \frac{(\frac{1}{\sqrt{2}})^{2}}{(\frac{\sqrt{3}}{2})^{2}} + \frac{(\frac{1}{2})^{2}}{(\frac{1}{\sqrt{2}})^{2}} - \frac{(\frac{1}{\sqrt{3}})^{2}}{(1)^{2}} - \frac{(\frac{1}{2})^{2}}{(\sqrt{3})^{2}}$$\Rightarrow \frac{1/2}{3/4} + \frac{1/4}{1/2} - \frac{1/3}{1} - \frac{1/4}{3}$$\Rightarrow (\frac{1}{2} 	imes \frac{4}{3}) + (\frac{1}{4} 	imes 2) - \frac{1}{3} - \frac{1}{12}$$\Rightarrow \frac{2}{3} + \frac{1}{2} - \frac{1}{3} - \frac{1}{12}$Taking $LCM$ as $12$: $\frac{8 + 6 - 4 - 1}{12} = \frac{9}{12} = \frac{3}{4}$.

The numerical value of $\frac{\cos ^{2} 45^{\circ}}{\sin ^{2} 60^{\circ}}+\frac{\cos ^{2} 60^{\circ}}{\sin ^{2} 45^{\circ}}-\frac{\tan ^{2} 30^{\circ}}{\cot ^{2} 45^{\circ}}-\frac{\sin ^{2} 30^{\circ}}{\cot ^{2} 30^{\circ}}$ is

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