Question

The value of \( \frac{1}{\sin 10^\circ} - \frac{\sqrt{3}}{\cos 10^\circ} \) is equal to

• A. 4
• B. 3
• C. 2
• D. 1

Hint:

When dealing with trigonometric expressions involving specific angles, try to relate them to standard angles (like 30°, 45°, 60°) and use sum/difference and multiple angle formulas.

Answer 1

The Correct Option is A

Step 1: Combine the fractions.
$$\frac{1}{\sin 10^\circ} - \frac{\sqrt{3}}{\cos 10^\circ} = \frac{\cos 10^\circ - \sqrt{3} \sin 10^\circ}{\sin 10^\circ \cos 10^\circ}$$ Step 2: Manipulate the numerator to use trigonometric identities.
We can divide and multiply the numerator by 2:
$$\frac{2 \left( \frac{1}{2} \cos 10^\circ - \frac{\sqrt{3}}{2} \sin 10^\circ \right)}{\sin 10^\circ \cos 10^\circ}$$ We know that \( \cos 60^\circ = \frac{1}{2} \) and \( \sin 60^\circ = \frac{\sqrt{3}}{2} \). Substituting these values:
$$\frac{2 \left( \cos 60^\circ \cos 10^\circ - \sin 60^\circ \sin 10^\circ \right)}{\sin 10^\circ \cos 10^\circ}$$ Step 3: Apply the cosine addition formula.
The formula is \( \cos(A + B) = \cos A \cos B - \sin A \sin B \). Here, \( A = 60^\circ \) and \( B = 10^\circ \):
$$\frac{2 \cos(60^\circ + 10^\circ)}{\sin 10^\circ \cos 10^\circ} = \frac{2 \cos 70^\circ}{\sin 10^\circ \cos 10^\circ}$$ Step 4: Use the identity \( \cos \theta = \sin(90^\circ - \theta) \).
$$\cos 70^\circ = \sin(90^\circ - 70^\circ) = \sin 20^\circ$$ So the expression becomes:
$$\frac{2 \sin 20^\circ}{\sin 10^\circ \cos 10^\circ}$$ Step 5: Use the double angle formula for sine.
The formula is \( \sin 2\theta = 2 \sin \theta \cos \theta \). Here, \( \theta = 10^\circ \), so \( \sin 20^\circ = 2 \sin 10^\circ \cos 10^\circ \). Substituting this into the expression:
$$\frac{2 (2 \sin 10^\circ \cos 10^\circ)}{\sin 10^\circ \cos 10^\circ}$$ Step 6: Cancel out common terms.
$$\frac{4 \sin 10^\circ \cos 10^\circ}{\sin 10^\circ \cos 10^\circ} = 4$$ Thus, the value of the expression is 4.