Question

The value of \( \tan 15^\circ + \tan 75^\circ \) is equal to

• A. \(2\sqrt{3}\)
• B. \(2\)
• C. \(2-\sqrt{3}\)
• D. \(4\sqrt{3}\)
• E. \(4\)

Hint:

Use complementary angle identities to simplify expressions like \( \tan 75^\circ \).

Answer 1

Answer: (E)

Concept: Use identity: \[ \tan(90^\circ - \theta) = \cot \theta \]

Step 1: Convert
\[ \tan 75^\circ = \cot 15^\circ = \frac{1}{\tan 15^\circ} \]

Step 2: Let \( t = \tan 15^\circ \)
\[ \Rightarrow \tan 75^\circ = \frac{1}{t} \]

Step 3: Add
\[ t + \frac{1}{t} \]

Step 4: Use identity
\[ t = 2 - \sqrt{3} \] \[ \frac{1}{t} = 2 + \sqrt{3} \]

Step 5: Sum
\[ (2 - \sqrt{3}) + (2 + \sqrt{3}) = 4 \]

Step 6: Final Answer
\[ \boxed{4} \]