Breaking News: 0 = 4
Consider this proof that 4 = 0. Since cos^{2}(x) = 1  sin^{2}(x), we take the square roots of both sides to get an expression for cos (x). Then, add 1 to both sides, 1+cos (x) = 1 + [1  sin^{2}(x)]^{1/2}. Now, square both sides, [1+cos (x)]^{2} = [1 + [1  sin^{2}(x)]^{1/2}]^{2}. Since the two sides are equivalent, we substitute x= PI into each side to get [1+cos(PI)]^{2} = [1 + [1  sin^{2}(PI)]^{1/2}]^{2}, simplified is : [1+(1)]^{2} = [1+(10)^{1/2}]^{2}. Or, 0 = (1+1)^{2} = 4
What is wrong with this “proof”? Or is everything correct and 0 does equal 4?
Hint: First, 0 does not equal 4, thus an error is involved somewhere, even though everything seems to check out stepbystep.
Second, the key lies in a fundamental property "covered" in an algebra class, though the implications of its misuse often are not "uncovered."
Solution Commentary: Using a graphical calculator, graph the two sides of each equation in the proof, starting with cos^{2}(x) = 1  sin^{2}(x).
When the graphs for the two sides do not match, an error is signaled...now, focus in on what happened in the last step taken to get to that equation. What could be wrong?
Another approach is to substitute a wide range of values other than PI in the expression......look for a pattern in what values work and which do not work.
And if all else fails, think...In algebra, what is the equivalent expression for SQRT(x^{2})?
