Page 40 - Calculus for the Clueless, Calc II

The sharp-eyed reader will discover there are many, many other ways to do this problem. When a publisher
becomes smart enough to publish this book, this problem will become a contest.

Here is the weirdest miscellaneous item of all! If we have sin x or cos x in the denominator, and if they are
added or subtracted to each other (with one being multiplied by a number or being added or subtracted by itself
from a number), then the substitution is u = tan (x/2)!!!!!!!! I don't know who discovered it or why, but it works.

Let us derive all the parts. Otherwise you would never believe it. Let tan (x/2) = u = u/1. Draw the triangle for
2 1/2
2 1/2
x/2. We get sin (x/2) = u/(1 + u ) and cos (x/2)= 1/(1 + u ) .

Draw triangle for angle x.

Finally if u = tan (x/2), then tan u = x/2. Taking differentials, we get
-1

In summary,