Question

The value of \( \frac{\cos 20^\circ \cos 70^\circ - \sin 20^\circ}{\sin 70^\circ} \) is:

• A. \( \infty \)
• B. None of these
• C. 1
• D. 0

Hint:

For trigonometric simplifications, remember key identities such as \( \cos(90^\circ - x) = \sin x \) and \( \sin(90^\circ - x) = \cos x \) to help simplify expressions.

Answer 1

The Correct Option is D

Using the identity \( \cos (90^\circ - x) = \sin x \), we have: \[ \cos 70^\circ = \sin 20^\circ \quad \text{and} \quad \sin 70^\circ = \cos 20^\circ \] Substituting: \[ \frac{\cos 20^\circ \cos 70^\circ - \sin 20^\circ}{\sin 70^\circ} = \frac{\cos 20^\circ \sin 20^\circ - \sin 20^\circ}{\cos 20^\circ} \] Factoring \( \sin 20^\circ \): \[ \frac{\sin 20^\circ (\cos 20^\circ - 1)}{\cos 20^\circ} \] Since \( \cos 20^\circ - 1 = 0 \), the expression simplifies to: \[ 0 \] Therefore, the answer is 0.