Question

The value of $\frac{\tan 75^{\circ}+\tan 15^{\circ}}{\tan 75^{\circ}-\tan 15^{\circ}}$ is equal to

• A. $\frac{1}{\sqrt{3}}$
• B. $\sqrt{3}$
• C. 1
• D. $-\sqrt{3}$
• E. $\frac{2}{\sqrt{3}}$

Hint:

Logic Tip: Another approach is substituting the known exact values: $\tan 75^{\circ} = 2 + \sqrt{3}$ and $\tan 15^{\circ} = 2 - \sqrt{3}$. Numerator: $(2 + \sqrt{3}) + (2 - \sqrt{3}) = 4$. Denominator: $(2 + \sqrt{3}) - (2 - \sqrt{3}) = 2\sqrt{3}$. Result: $\frac{4}{2\sqrt{3}} = \frac{2}{\sqrt{3}}$.

Answer 1

Answer: (E)

Concept:
Converting tangent expressions to sines and cosines is a highly effective strategy for simplification. Recall that $\tan x = \frac{\sin x}{\cos x}$. By substituting this, we can utilize the compound angle formulas for sine: $\sin(A + B) = \sin A\cos B + \cos A\sin B$ $\sin(A - B) = \sin A\cos B - \cos A\sin B$
Step 1: Rewrite the expression in terms of sine and cosine.
$$\frac{\frac{\sin 75^{\circ}}{\cos 75^{\circ}} + \frac{\sin 15^{\circ}}{\cos 15^{\circ}}}{\frac{\sin 75^{\circ}}{\cos 75^{\circ}} - \frac{\sin 15^{\circ}}{\cos 15^{\circ}}}$$
Step 2: Find a common denominator for the numerator and denominator.
Multiply the top terms and bottom terms by the common denominator $(\cos 75^{\circ}\cos 15^{\circ})$: $$\text{Numerator} = \sin 75^{\circ}\cos 15^{\circ} + \cos 75^{\circ}\sin 15^{\circ}$$ $$\text{Denominator} = \sin 75^{\circ}\cos 15^{\circ} - \cos 75^{\circ}\sin 15^{\circ}$$
Step 3: Apply sine compound angle formulas.
Notice that the numerator perfectly matches the expansion of $\sin(A + B)$ and the denominator matches $\sin(A - B)$, where $A = 75^{\circ}$ and $B = 15^{\circ}$. $$\text{Numerator} = \sin(75^{\circ} + 15^{\circ}) = \sin(90^{\circ}) = 1$$ $$\text{Denominator} = \sin(75^{\circ} - 15^{\circ}) = \sin(60^{\circ}) = \frac{\sqrt{3}}{2}$$
Step 4: Evaluate the final fraction.
$$\text{Value} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}}$$