Perhaps your math education is just shallow

I came across this interesting book review here: http://blogs.scientificamerican.com/roots-of-unity/2014/05/31/how-not-to-be-wrong-book-review/

This chart here captures my attention:

All the math that most people ever had, including that complex looking integral on lower right are shallow stuff. This means, most of you haven't seen any profound things yet. Those on the upper right are difficult and profound. I guess I have to read the book to see what's simple and profound.

Now I have something to say about sin 2x = 2 sin x cos x.

Seen this? All high school students should have. This follows immediately from the Addition Formula: sin (x + y) = sin x cos y + cos y sin x. Remember! sin (x + y) is not sin x + sin y! Don't believe me? Compare sin 90° vs sin 30° + sin 60°. (I specifically chose choices so that you don't need a calculator) Grab a scientific calculator if you don't even remember your special triangles.

Now the proof of the Addition Formula is actually not so straight forward. Draw a right triangle, split an acute angle to x and y.. and assume you know sin x and sin y. Trying to determine sin (x + y) (which is opposite over hypotenuse of that triangle you just drew) is going to be over-your-head for vast majority of people....even for many sharp high school students. Now even if you get that... what if your angle is greater than 90 degrees? Ok, I assure you the Addition Formula still holds for every real x and y. This is not so straightforward. That is why the proof should be in your book. This is why this formula deserves a box around and even worth remembering (I did since high school). Sure, look it up on the web if you're interested.

The author is putting this in same category as 1 + 2 = 3. I would move that to higher up in the shallow-profound axis.

For that integral in lower right... to evaluate that my first intuition would be u-substitution. Alas, does not work. Probably need to try by-part or trig-substitution (and I won't bother actually trying to evaluate this problem from thin air). So yes this is complicated... but still shallow. Most people's entire math education is just shallow... unless you are in graduate school or a professor.

I probably won't ever get out of the "shallow" quadrants... and that's totally fine. At least I understand my shallow topics.