That "double" less than symbol is used to say "much smaller than". It is of course a qualitative symbol, and it has no rigorous definition.
The exercise asks you to verify, numerically, that if you choose a very, very small input, then the approximations are good. Take the first one for example: we claim that if we pick a very small angle, its cosine is almost one. Let's choose [tex] \theta = 0.00001 [/tex]. If we compute its cosine, we have
[tex] \cos(0.00001) = 0.99999999995 [/tex]
So, approximating the result with 1 is a good approximation.
Similarly, in the second example, we claim that for very small angles, the sine of the angle is almost the angle itself: choosing for example [tex] \theta = 0.001[/tex], we have
[tex] \sin(0.00001) = 0.00099999983333\ldots [/tex]
So the output is almost identically to the input.
For the third, we claim that [tex] e^{-x} [/tex] and [tex]1-x [/tex] are almost the same thing if x is very small. So, for example, we pick [tex] x = 0.00001 [/tex] and we have
[tex] e^{-0.00001} = 0.99999000005\ldots[/tex]
whereas
[tex] 1 - 0.00001 = 0.99999 [/tex]
so the two results are almost identical.
You can keep going like this and "prove" all the remaining approximations
