What is `cos80^@ + cos 40^@ - cos20^@` equal to ?

Options

A. 2
B. 1
C. 0
D. `-19`

Answer

To solve the expression \( \cos 80^\circ + \cos 40^\circ - \cos 20^\circ \), we can follow these steps:

### Step 1: Use the Cosine Addition Formula
We can use the formula for the sum of cosines:
\[ \cos C + \cos D = 2 \cos\left(\frac{C + D}{2}\right) \cos\left(\frac{C - D}{2}\right) \]
In our case, let \( C = 80^\circ \) and \( D = 40^\circ \).

### Step 2: Apply the Formula
Substituting \( C \) and \( D \) into the formula:
\[ \cos 80^\circ + \cos 40^\circ = 2 \cos\left(\frac{80^\circ + 40^\circ}{2}\right) \cos\left(\frac{80^\circ - 40^\circ}{2}\right) \]
Calculating the angles:
\[ \frac{80^\circ + 40^\circ}{2} = \frac{120^\circ}{2} = 60^\circ \]
\[ \frac{80^\circ - 40^\circ}{2} = \frac{40^\circ}{2} = 20^\circ \]
Thus, we have:
\[ \cos 80^\circ + \cos 40^\circ = 2 \cos 60^\circ \cos 20^\circ \]

### Step 3: Substitute the Value of \( \cos 60^\circ \)
We know that \( \cos 60^\circ = \frac{1}{2} \):
\[ \cos 80^\circ + \cos 40^\circ = 2 \cdot \frac{1}{2} \cdot \cos 20^\circ = \cos 20^\circ \]

### Step 4: Substitute Back into the Original Expression
Now, substituting back into the original expression:
\[ \cos 80^\circ + \cos 40^\circ - \cos 20^\circ = \cos 20^\circ - \cos 20^\circ = 0 \]

### Conclusion
Thus, the final result is:
\[ \cos 80^\circ + \cos 40^\circ - \cos 20^\circ = 0 \]

### Answer
The answer is \( 0 \).

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