Show that:

cos 24° + cos 55° + cos 125° + cos 204° + cos 300° = ½

To prove that \( \cos 24^\circ + \cos 55^\circ + \cos 125^\circ + \cos 204^\circ + \cos 300^\circ = \frac{1}{2} \), we can use the properties of cosine and some trigonometric identities.

Step-by-Step Breakdown

First, let's group the angles based on their symmetry:

• Notice that \( \cos 125^\circ \) and \( \cos 204^\circ \) can be paired with \( \cos 55^\circ \) and \( \cos 24^\circ \).
• Also, \( \cos 300^\circ \) can be rewritten as \( \cos(360^\circ - 60^\circ) = \cos 60^\circ = \frac{1}{2} \).

Calculating Each Cosine

Now, let's calculate the cosines:

• Using the identity \( \cos(180^\circ - x) = -\cos x \):
• We find that \( \cos 125^\circ = -\cos 55^\circ \) and \( \cos 204^\circ = -\cos 24^\circ \).

Combining the Terms

Now, we can rewrite the original expression:

\( \cos 24^\circ + \cos 55^\circ + (-\cos 55^\circ) + (-\cos 24^\circ) + \frac{1}{2} \)

This simplifies to:

\( 0 + \frac{1}{2} = \frac{1}{2} \)

Final Result

Thus, we have shown that:

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