An elegant example of such mathematical fallacy is good to keep an idiot busy for couple of days. The one I’ll describe next is just what you need.

It starts from the cos²x + sin²x = 1 which is the trigonometric form of the Pythagorean theorem, and it goes like this:

cos²x + sin²x = 1
cos²x = 1 – sin²x
cos x = (1 – sin²x)^1/2
1 + cos x = 1 + (1 – sin²x)^1/2
(1 + cos x)² = (1 + (1 – sin²x)^1/2)²
for x = π (pi = 3,14159265…), we have,
(1 – 1)² = (1 + (1 – 0)^1/2)²
0 = (1 + 1)²
0 = 4 //interesting ?!

The mistake is a little hard to notice, but the result should tell you that something is wrong.