One life-lesson from math is that sometimes you can solve a problem without doing what the problem at first seems to require. I’ll give an elementary example and a more advanced example.
The first example is finding remainders. What is the remainder when 5,000,070,004 is divided by 9? At first it may seem that you need to divide 5,000,070,004 by 9, but you don’t. You weren’t asked the quotient, only the remainder, which in this case you can do directly. By casting out nines, you can quickly see the remainder is 7.
The second example is definite integrals. The usual procedure for computing definite integrals is to first find an indefinite integral (i.e. anti-derivative) and take the difference of its values at the two end points. But sometimes it’s possible to find the definite integral directly, even when you couldn’t first find the indefinite integral. Maybe you can evaluate the definite integral by symmetry, or a probability argument, or by contour integration, or some other trick.
Contour integration is an interesting example because you don’t do what you might think you need to — i.e. find an indefinite integral — but you do have to do something you might never imagine doing before you’ve seen the trick, i.e. convert an integral over the real line to an integral in the complex plane to make it simpler!
What are some more examples, mathematical or not, of solving a problem without doing something that at first seems necessary?