Question

Find the value of $\cos 24^\circ+\cos 55^\circ+\cos 125^\circ+\cos 156^\circ$.

• A. $0$
• B. $2$
• C. $3$
• D. $1$

Hint:

Whenever you see angles that are supplements ($\theta$ and $180^\circ-\theta$), pair them using $\cos(180^\circ-\theta)=-\cos\theta$ to simplify quickly.

Answer 1

The Correct Option is A

Step 1: Pair adjacent terms with the sum-to-product identity: \( \cos24^\circ+\cos55^\circ=2\cos(39.5^\circ)\cos(15.5^\circ) \) and \( \cos125^\circ+\cos156^\circ=2\cos(140.5^\circ)\cos(15.5^\circ) \).

Step 2: Since \( \cos(140.5^\circ)=\cos(180^\circ-39.5^\circ)=-\cos(39.5^\circ) \), the two products become \( 2\cos(15.5^\circ)\cos(39.5^\circ) \) and \( -2\cos(15.5^\circ)\cos(39.5^\circ) \).

Step 3: Adding them gives \( 0 \). \[ \boxed{0} \]