The Goal

Simplify the following:

The Mistake

Find the mistake:

(Roll the mouse over the math to see a hint in red)

The Correction

(Roll the mouse over the area above to see the correction in blue)

An Explanation

This is an example of awful cancellation that results from not understanding the function notation of trigonometric functions. Cosine always comes with an argument, that is, an input. Cosine is a function! Think about the square root function. We say √x as "square root of x", not "square root times x", for a good reason. We plug x into the square root function. It is meaningless to talk about multiplying the square root symbol by anything. In the same way, when we write cos(2x) we mean 2x plugged into the cosine function. Cosine by itself has no meaning without an argument! And since the "cos" does not represent a factor in the expression of this example, it cannot be cancelled.

It's not exactly clear how the mistake was made. Perhaps the first mistake was misinterpreting cos(2x) as cos²(x), which then led to the bogus cancellation of a "cos" and an x from both numerator and denominator. Neither cancellation is legal.