Evaluate : `cos24^(@)+cos55^(@)+cos125^(@)+cos204^(@)+cos300^(@)`.

Answer

To evaluate the expression \( \cos 24^\circ + \cos 55^\circ + \cos 125^\circ + \cos 204^\circ + \cos 300^\circ \), we can use the properties of cosine and the unit circle. Here’s a step-by-step solution:

### Step 1: Rewrite the cosine terms using angle identities
We can use the identity \( \cos(180^\circ - x) = -\cos x \) and \( \cos(180^\circ + x) = -\cos x \) to rewrite some of the terms.

- \( \cos 125^\circ = \cos(180^\circ - 55^\circ) = -\cos 55^\circ \)
- \( \cos 204^\circ = \cos(180^\circ + 24^\circ) = -\cos 24^\circ \)

### Step 2: Substitute the rewritten terms into the expression
Now we can substitute these identities into the original expression:

\[ \cos 24^\circ + \cos 55^\circ + \cos 125^\circ + \cos 204^\circ + \cos 300^\circ \]

becomes:

\[ \cos 24^\circ + \cos 55^\circ - \cos 55^\circ - \cos 24^\circ + \cos 300^\circ \]

### Step 3: Simplify the expression
Notice that \( \cos 24^\circ \) and \( -\cos 24^\circ \) cancel each other out, and \( \cos 55^\circ \) and \( -\cos 55^\circ \) also cancel out:

\[ 0 + \cos 300^\circ \]

### Step 4: Evaluate \( \cos 300^\circ \)
Now we need to evaluate \( \cos 300^\circ \). Since \( 300^\circ \) is in the fourth quadrant, we can use the identity:

\[ \cos 300^\circ = \cos(360^\circ - 60^\circ) = \cos 60^\circ = \frac{1}{2} \]

### Final Answer
Thus, the final result is:

\[ \cos 24^\circ + \cos 55^\circ + \cos 125^\circ + \cos 204^\circ + \cos 300^\circ = \frac{1}{2} \]