Mystery Math: 0 Equals to 2

Monday, 30 April 2012

Consider the equation

$\cos^2x=1-\sin^2x$

which holds as a consequence of the Pythagorean theorem. Then, by taking a square root,

$\cos x = (1-\sin^2x)^{1/2}$

so that

$1+\cos x = 1+(1-\sin^2x)^{1/2}.$

But evaluating this when x = π implies

$1-1 = 1+(1-0)^{1/2}$

$0=2$

which is absurd.

The error in each of these examples fundamentally lies in the fact that any equation of the form

$x^2 = a^2$

has two solutions, provided a ≠ 0,

$x=\pm a$

Here , the fallacy has occurred .

Monday, 30 April 2012