Use Differentials to Approximate Csc 61

In this tutorial we shall develop the differentials approximate the value of $\csc {61^ \circ }$.
The nearest number to 61 degree whose cosecant value can be taken is 60 degree, so let us consider that $x = {60^ \circ }$ and $dx = {1^ \circ } = \frac{\pi }{{180}} = 0.0174$.
Now consider

Differentiate equation (i) with respect to $x$, we have

Taking the differential of equation (ii), we get

Putting the values $x = {60^ \circ }$ and $dx = 0.0174$, we have

Now