`cos 24^(@)+cos5^(@)+ cos 175^(@) + cos 204^(@) + cos 300^(@) =`

Options

A. `(1)/(2)`
B. `- ( 1)/( 2)`
C. `(sqrt(3))/(2)`
D. 1

Answer

To solve the equation \( \cos 24^\circ + \cos 5^\circ + \cos 175^\circ + \cos 204^\circ + \cos 300^\circ \), we will simplify each cosine term step by step.

### Step 1: Simplify \( \cos 175^\circ \)
Using the identity \( \cos(180^\circ - \theta) = -\cos(\theta) \):
\[ \cos 175^\circ = \cos(180^\circ - 5^\circ) = -\cos 5^\circ \]

### Step 2: Simplify \( \cos 204^\circ \)
Using the identity \( \cos(180^\circ + \theta) = -\cos(\theta) \):
\[ \cos 204^\circ = \cos(180^\circ + 24^\circ) = -\cos 24^\circ \]

### Step 3: Simplify \( \cos 300^\circ \)
Using the identity \( \cos(360^\circ - \theta) = \cos(\theta) \):
\[ \cos 300^\circ = \cos(360^\circ - 60^\circ) = \cos 60^\circ = \sin 30^\circ \]

### Step 4: Substitute the simplified values into the equation
Now substituting the simplified values back into the equation:
\[ \cos 24^\circ + \cos 5^\circ + (-\cos 5^\circ) + (-\cos 24^\circ) + \sin 30^\circ \]

### Step 5: Combine like terms
Notice that \( \cos 24^\circ \) and \( -\cos 24^\circ \) cancel each other out, and \( \cos 5^\circ \) and \( -\cos 5^\circ \) also cancel each other out:
\[ 0 + 0 + \sin 30^\circ = \sin 30^\circ \]

### Step 6: Evaluate \( \sin 30^\circ \)
We know that:
\[ \sin 30^\circ = \frac{1}{2} \]

### Final Result
Thus, the value of the expression is:
\[ \cos 24^\circ + \cos 5^\circ + \cos 175^\circ + \cos 204^\circ + \cos 300^\circ = \frac{1}{2} \]

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