Math Problem 7: Proof that cos(pi/4) = 1
Remember that sin(pi/4) = cos(pi/4), tan(pi/4) = 1, and tan(x) = sin(x) / cos(x).
Starting from the obviously true equation 1 + sin(pi/4) - cos(pi/4) = tan(pi/4), substitite tan(pi/4) = sin(pi/4) / cos(pi/4) to get 1 + sin(pi/4) - cos(pi/4) = sin(pi/4) / cos(pi/4). Multiply every term by cos(pi/4) to get cos(pi/4) + sin(pi/4) * cos(pi/4) - cos(pi/4) * cos(pi/4) = sin(pi/4). Subtract a cos(pi/4) from each side and factor out cos(pi/4) from the left-hand side and you'll get cos(pi/4) * [ sin(pi/4) - cos(pi/4) ] = sin (pi/4) - cos(pi/4). Divide both sides by sin(pi/4) - cos(pi/4) and you get the solution cos(pi/4) = 1.
But cos(pi/4) does not equal 1. What is wrong?
One of the operations performed results in an undefined value.
The problem is with the final division. sin(pi/4) - cos(pi/4) is 0, and division by zero is not a legal mathematical operation; the results are undefined.