Evaluate each of the following

Solution:

We have,

$\frac{\sin 30^{\circ}}{\sin 45^{\circ}}+\frac{\tan 45^{\circ}}{\sec 60^{\circ}}-\frac{\sin 60^{\circ}}{\cot 45^{\circ}}-\frac{\cos 30^{\circ}}{\sin 90^{\circ}}$....(1)

Now,

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

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

So by substituting above values in equation (1)

We get,

$\frac{\sin 30^{\circ}}{\sin 45^{\circ}}+\frac{\tan 45^{\circ}}{\sec 60^{\circ}}-\frac{\sin 60^{\circ}}{\cot 45^{\circ}}-\frac{\cos 30^{\circ}}{\sin 90^{\circ}}$

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

Now by further simplifying

We get,

$\frac{\sin 30^{\circ}}{\sin 45^{\circ}}+\frac{\tan 45^{\circ}}{\sec 60^{\circ}}-\frac{\sin 60^{\circ}}{\cot 45^{\circ}}-\frac{\cos 30^{\circ}}{\sin 90^{\circ}}$

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

Since, $2=\sqrt{2} \times \sqrt{2}$

Therefore,

$\frac{\sin 30^{\circ}}{\sin 45^{\circ}}+\frac{\tan 45^{\circ}}{\sec 60^{\circ}}-\frac{\sin 60^{\circ}}{\cot 45^{\circ}}-\frac{\cos 30^{\circ}}{\sin 90^{\circ}}$

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

Now, one $\sqrt{2}$ gets cancelled and

We get,

$\frac{\sin 30^{\circ}}{\sin 45^{\circ}}+\frac{\tan 45^{\circ}}{\sec 60^{\circ}}-\frac{\sin 60^{\circ}}{\cot 45^{\circ}}-\frac{\cos 30^{\circ}}{\sin 90^{\circ}}$

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

Now, by taking LCM

We get,

$\frac{\sin 30^{\circ}}{\sin 45^{\circ}}+\frac{\tan 45^{\circ}}{\sec 60^{\circ}}-\frac{\sin 60^{\circ}}{\cot 45^{\circ}}-\frac{\cos 30^{\circ}}{\sin 90^{\circ}}$

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

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

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

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

Therefore,

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