What is the value of cos `20^@` + cos `22^@` + cos `158^@` + cos `200^@` + cos `300^@` ?

Answer

To find the value of \( \cos 20^\circ + \cos 22^\circ + \cos 158^\circ + \cos 200^\circ + \cos 300^\circ \), we can simplify the expression step by step.

### Step 1: Rewrite the angles
We can rewrite some of the angles using trigonometric identities:
- \( \cos 158^\circ = \cos(180^\circ - 22^\circ) = -\cos 22^\circ \)
- \( \cos 200^\circ = \cos(180^\circ + 20^\circ) = -\cos 20^\circ \)
- \( \cos 300^\circ = \cos(360^\circ - 60^\circ) = \cos 60^\circ \)

### Step 2: Substitute the rewritten angles into the expression
Now we substitute these values back into the expression:
\[ \cos 20^\circ + \cos 22^\circ + (-\cos 22^\circ) + (-\cos 20^\circ) + \cos 60^\circ \]

### Step 3: Simplify the expression
Now we can simplify the expression:
\[ \cos 20^\circ - \cos 20^\circ + \cos 22^\circ - \cos 22^\circ + \cos 60^\circ \]
This simplifies to:
\[ 0 + 0 + \cos 60^\circ = \cos 60^\circ \]

### Step 4: Find the value of \( \cos 60^\circ \)
We know that:
\[ \cos 60^\circ = \frac{1}{2} \]

### Final Answer
Thus, the value of \( \cos 20^\circ + \cos 22^\circ + \cos 158^\circ + \cos 200^\circ + \cos 300^\circ \) is:
\[ \frac{1}{2} \]