Question

Evaluate the following expression: $ \frac{\cos 75^\circ - \cos 15^\circ}{\cos 75^\circ + \cos 15^\circ} $

• A. \( 0 \)
• B. \( \frac{1}{2} \)
• C. \( 1 \)
• D. \( -1 \)

Hint:

Use sum and difference trigonometric identities to simplify expressions involving cosines.

Answer 1

The Correct Option is A

We are asked to evaluate the following expression: \[ \frac{\cos 75^\circ - \cos 15^\circ}{\cos 75^\circ + \cos 15^\circ} \] We can use the trigonometric identity for the sum and difference of cosines: \[ \cos A - \cos B = -2 \sin\left( \frac{A+B}{2} \right) \sin\left( \frac{A-B}{2} \right) \] \[ \cos A + \cos B = 2 \cos\left( \frac{A+B}{2} \right) \cos\left( \frac{A-B}{2} \right) \] Using these identities, we get: \[ \cos 75^\circ - \cos 15^\circ = -2 \sin\left( \frac{75^\circ + 15^\circ}{2} \right) \sin\left( \frac{75^\circ - 15^\circ}{2} \right) = -2 \sin(45^\circ) \sin(30^\circ) \] \[ \cos 75^\circ + \cos 15^\circ = 2 \cos\left( \frac{75^\circ + 15^\circ}{2} \right) \cos\left( \frac{75^\circ - 15^\circ}{2} \right) = 2 \cos(45^\circ) \cos(30^\circ) \] Now, substituting these values into the original expression: \[ \frac{-2 \sin(45^\circ) \sin(30^\circ)}{2 \cos(45^\circ) \cos(30^\circ)} = \frac{- \sin(45^\circ) \sin(30^\circ)}{\cos(45^\circ) \cos(30^\circ)} \] Since \( \sin(45^\circ) = \cos(45^\circ) \) and simplifying: \[ \frac{- \sin(30^\circ)}{\cos(30^\circ)} = - \tan(30^\circ) = 0 \]
Thus, the value of the expression is \( 0 \).