Question:
$\frac{\cos 75^{\circ} - \cos 15^{\circ}}{\cos 75^{\circ} + \cos 15^{\circ}} =$
$\frac{\cos 75^{\circ} - \cos 15^{\circ}}{\cos 75^{\circ} + \cos 15^{\circ}} =$
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$\frac{\cos A - \cos B}{\cos A + \cos B} = -\tan(\frac{A+B}{2}) \tan(\frac{A-B}{2})$.
Updated On: Apr 28, 2026
- $\frac{-1}{\sqrt{3}}$
- $\frac{1}{\sqrt{2}}$
- $\frac{1}{\sqrt{3}}$
- $\frac{-1}{\sqrt{2}}$
- $\sqrt{3}$
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The Correct Option is A
Solution and Explanation
Step 1: Concept
Use the sum-to-product formulas.
Step 2: Analysis
Numerator: $\cos 75^{\circ} - \cos 15^{\circ} = -2 \sin 45^{\circ} \sin 30^{\circ}$. Denominator: $\cos 75^{\circ} + \cos 15^{\circ} = 2 \cos 45^{\circ} \cos 30^{\circ}$.
Step 3: Calculation
Value $= \frac{-2 \sin 45^{\circ} \sin 30^{\circ}}{2 \cos 45^{\circ} \cos 30^{\circ}} = -\tan 45^{\circ} \tan 30^{\circ}$. Value $= -1 \cdot \frac{1}{\sqrt{3}} = -\frac{1}{\sqrt{3}}$. Final Answer: (A)
Use the sum-to-product formulas.
Step 2: Analysis
Numerator: $\cos 75^{\circ} - \cos 15^{\circ} = -2 \sin 45^{\circ} \sin 30^{\circ}$. Denominator: $\cos 75^{\circ} + \cos 15^{\circ} = 2 \cos 45^{\circ} \cos 30^{\circ}$.
Step 3: Calculation
Value $= \frac{-2 \sin 45^{\circ} \sin 30^{\circ}}{2 \cos 45^{\circ} \cos 30^{\circ}} = -\tan 45^{\circ} \tan 30^{\circ}$. Value $= -1 \cdot \frac{1}{\sqrt{3}} = -\frac{1}{\sqrt{3}}$. Final Answer: (A)
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