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,

$sin 30^{\circ}=\frac{1}{2}$

$\sin 45^{\circ}=\frac{1}{\sqrt2}$

$\tan 45^{\circ}=1$

$\sec 60^{\circ}=2$

$\sin 60^{\circ}=\frac{\sqrt3}{2}$

$\cot 45^{\circ}=1$

$\cos 30^{\circ}=\frac{\sqrt3}{2}$

$\sin 90^{\circ}=1$

$\frac{\sin 30^{\circ}}{\sin 45^{\circ}}+\frac{\tan 45^{\circ}}{\sec 60^{\circ}}-\frac{\sin 60^{\circ}}{\cot 45^{\circ}}-\frac{\cos 30^{\circ}}{\sin 90^{\circ}}=\frac{\frac{1}{2}}{\frac{1}{\sqrt{2}}} +\frac{1}{2} -\left(\frac{\frac{\sqrt{3}}{2}}{1}\right) -\left(\frac{\frac{\sqrt{3}}{2}}{1}\right)$

$=\frac{1}{2} \times \frac{\sqrt{2}}{1} +\frac{1}{2} -\frac{\sqrt{3}}{2} -\frac{\sqrt{3}}{2}$

$=\frac{\sqrt{2} +1-\sqrt{3} -\sqrt{3}}{2}$

$=\frac{\sqrt{2} +1-2\sqrt{3}}{2}$

Hence, $\frac{\sin 30^{\circ}}{\sin 45^{\circ}}+\frac{\tan 45^{\circ}}{\sec 60^{\circ}}-\frac{\sin 60^{\circ}}{\cot 45^{\circ}}-\frac{\cos 30^{\circ}}{\sin 90^{\circ}}=\frac{\sqrt{2} +1-2\sqrt{3}}{2}$.   

Tutorialspoint

Updated on: 10-Oct-2022

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