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 the student attempts to clear the denominators by introducing the sin(x) and cos(x) factors, completely incorrectly. The second step is a complete mystery.

In the second attempted solution coefficient 2 is missing from the cosine double-angle formula.

In the third mistake cosine double-angle formula is incorrect.

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