Some problems provide the opportunity for more than one mistake.

The Goal

Simplify the expression:

The Mistakes

Find the mistakes:

1.

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

2.

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

3.

(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)

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.