Question

The value of \( \cot 70^\circ + 4\cos 70^\circ \) is:

• A. $\sqrt{3}$
• B. $2\sqrt{3}$
• C. $1/2$
• D. $\sqrt{3}/2$

Hint:

$\sin 2\theta = 2 \sin \theta \cos \theta$ is often the first step in simplifying such expressions.

Answer 1

The Correct Option is A

Step 1: Concept
Convert cotangent to sine and cosine and simplify using product-to-sum identities.
Step 2: Analysis
$\frac{\cos 70^{\circ}}{\sin 70^{\circ}} + 4 \cos 70^{\circ} = \frac{\cos 70^{\circ} + 4 \sin 70^{\circ} \cos 70^{\circ}}{\sin 70^{\circ}} = \frac{\cos 70^{\circ} + 2 \sin 140^{\circ}}{\sin 70^{\circ}}$.
$\sin 140^{\circ} = \sin(180^{\circ}-40^{\circ}) = \sin 40^{\circ}$.
Expression becomes $\frac{\sin 20^{\circ} + 2 \sin 40^{\circ}}{\cos 20^{\circ}}$ (using complementary angles).
Step 3: Conclusion
Further simplification using $\sin A + \sin B$ formulas leads to $\tan 60^{\circ} = \sqrt{3}$.
Final Answer: (A)