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Evaluate each of the following:$ \frac{\sin 30^{\circ}}{\sin 45^{\circ}}+\frac{\tan 45^{\circ}}{\sec 60^{\circ}}-\frac{\sin 60^{\circ}}{\cot 45^{\circ}}-\frac{\cos 30^{\circ}}{\sin 90^{\circ}} $
Given:
\( \frac{\sin 30^{\circ}}{\sin 45^{\circ}}+\frac{\tan 45^{\circ}}{\sec 60^{\circ}}-\frac{\sin 60^{\circ}}{\cot 45^{\circ}}-\frac{\cos 30^{\circ}}{\sin 90^{\circ}} \)
To do:
We have to evaluate \( \frac{\sin 30^{\circ}}{\sin 45^{\circ}}+\frac{\tan 45^{\circ}}{\sec 60^{\circ}}-\frac{\sin 60^{\circ}}{\cot 45^{\circ}}-\frac{\cos 30^{\circ}}{\sin 90^{\circ}} \).
Solution:
We know that,
$sin 30^{\circ}=\frac{1}{2}$
$\sin 45^{\circ}=\frac{1}{\sqrt2}$
$\tan 45^{\circ}=1$
$\sec 60^{\circ}=2$
$\sin 60^{\circ}=\frac{\sqrt3}{2}$
$\cot 45^{\circ}=1$
$\cos 30^{\circ}=\frac{\sqrt3}{2}$
$\sin 90^{\circ}=1$
Therefore,$\frac{\sin 30^{\circ}}{\sin 45^{\circ}}+\frac{\tan 45^{\circ}}{\sec 60^{\circ}}-\frac{\sin 60^{\circ}}{\cot 45^{\circ}}-\frac{\cos 30^{\circ}}{\sin 90^{\circ}}=\frac{\frac{1}{2}}{\frac{1}{\sqrt{2}}} +\frac{1}{2} -\left(\frac{\frac{\sqrt{3}}{2}}{1}\right) -\left(\frac{\frac{\sqrt{3}}{2}}{1}\right)$
$=\frac{1}{2} \times \frac{\sqrt{2}}{1} +\frac{1}{2} -\frac{\sqrt{3}}{2} -\frac{\sqrt{3}}{2}$
$=\frac{\sqrt{2} +1-\sqrt{3} -\sqrt{3}}{2}$
$=\frac{\sqrt{2} +1-2\sqrt{3}}{2}$
Hence, $\frac{\sin 30^{\circ}}{\sin 45^{\circ}}+\frac{\tan 45^{\circ}}{\sec 60^{\circ}}-\frac{\sin 60^{\circ}}{\cot 45^{\circ}}-\frac{\cos 30^{\circ}}{\sin 90^{\circ}}=\frac{\sqrt{2} +1-2\sqrt{3}}{2}$.
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