Some problems provide the opportunity for more than one mistake.

The Goal

Solve the equation:

The Mistakes

Find the mistakes:

1.

Need a hint? Look carefully at the red part:

2.

Need a hint? Look carefully at the red part:

3.

Need a hint? Look carefully at the red part:

The Correction

Explanations

In the first mistake the coefficients of cos²(2x) and sin²(x) mysteriously vanished in the first step.

The second attempted solution makes use of an incorrect formula for cos(2x), in fact the negation of a correct formula.

In the third mistake once again an incorrect formula was used for cos(2x); recall that in fact cos²(x) + sin²(x) = 1, not cos(2x).

The keys to solving this problem are knowing the correct double angle formulas and then using correct algebra to solve the equation.