Some problems provide the opportunity for more than one mistake.

The Goal

Simplify the expression:

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 it appears that sin(2x) was interpreted as sin(x)sin(x), which is false; likewise for cos(2x). Sin(x) and cos(x) were then cancelled. The second step makes no sense, but perhaps was motivated by the expected answer (sec(x)).

In the second attempted solution the wrong formulas were used for sin(2x) and cos(2x); there seems to be confusion with the formulas for sin²(x) and cos²(x); even these are not correct. The second step is pure wishful thinking, but even that does not help.

In the third mistake the formula for cos(2x) is misrecalled, with the parentheses misplaced. The mistake is compounded with an incorrect cancellation in the second step - a denominator is simply erased.

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