The value of cos 20° + cos 100° + cos 140° - cos 140° - cos 200° is equal to

To solve the expression cos 20° + cos 100° + cos 140° - cos 140° - cos 200°, we can simplify it step by step.

Breaking Down the Expression

First, notice that cos 140° appears in both the positive and negative parts of the expression. This means it cancels out:

• cos 20° + cos 100° + cos 140° - cos 140° - cos 200° = cos 20° + cos 100° - cos 200°

Using Cosine Properties

Next, we can use the cosine addition formulas. Recall that:

• cos(180° - x) = -cos(x)

Applying this to cos 200°:

• cos 200° = cos(180° + 20°) = -cos 20°

Substituting Back

Now, we can substitute this back into our expression:

• cos 20° + cos 100° - (-cos 20°) = cos 20° + cos 100° + cos 20°

Simplifying Further

This simplifies to:

• 2cos 20° + cos 100°

Final Calculation

Next, we can use the identity for cos 100°:

• cos 100° = -sin 10°

Thus, our expression becomes:

• 2cos 20° - sin 10°

To find the exact value, you can use a calculator or trigonometric tables, but the simplified expression is:

2cos 20° - sin 10°