Question

Evaluate: \[ \frac{\cos 12^\circ - \sin 12^\circ}{\cos 12^\circ + \sin 12^\circ} + \frac{\sin 147^\circ}{\cos 147^\circ} \]

• A. -2
• B. 0
• C. -1
• D. 1

Hint:

For trigonometric expressions involving angles like these, try simplifying using standard identities such as \( \frac{\sin \theta}{\cos \theta} = \tan \theta \) and angle sum or difference identities.

Answer 1

The Correct Option is B

Step 1: Simplify the expression.
We can first simplify both parts of the expression separately. Consider the second term \( \frac{\sin 147^\circ}{\cos 147^\circ} \). Using the identity: \[ \frac{\sin \theta}{\cos \theta} = \tan \theta \] So, \[ \frac{\sin 147^\circ}{\cos 147^\circ} = \tan 147^\circ \] We know that \( 147^\circ = 180^\circ - 33^\circ \), and \( \tan(180^\circ - \theta) = -\tan \theta \), so: \[ \tan 147^\circ = -\tan 33^\circ \] Step 2: Simplify the first part of the expression.
For the first part, use the identity for simplifying the sum and difference of sine and cosine: \[ \frac{\cos 12^\circ - \sin 12^\circ}{\cos 12^\circ + \sin 12^\circ} = \frac{1 - \tan 12^\circ}{1 + \tan 12^\circ} \] which simplifies further using trigonometric identities. Step 3: Conclusion.
After simplifying both terms, the final answer is \( \boxed{0} \).