Originally Posted by

**paulrb** Are there any 1=2 "proofs" that do not involve dividing by zero?

Here is one.

$\displaystyle \cos^2x-\sin^2x\ =\ \cos2x$ (identity)

$\displaystyle \cos^2x\ =\ \cos2x+\sin^2x$ (rearramging)

$\displaystyle \cos x\ =\ \sqrt{\cos2x+\sin^2x}$ (taking square root)

$\displaystyle \cos\frac{3\pi}4\ =\ \sqrt{\cos\frac{3\pi}2+\sin^2\frac{3\pi}4}$ (substituing $\displaystyle x=\frac{3\pi}4)$

$\displaystyle -\frac1{\sqrt2}\ =\ \sqrt{0+\frac12}=\frac1{\sqrt2}$ (evaluating the substitution)

$\displaystyle -\frac12\ =\ \frac12$ (multiplying both sides by $\displaystyle \frac1{\sqrt2})$

$\displaystyle 1\ =\ 2$ (adding $\displaystyle \frac32$ to both sides)