Question

The value of tan 9°-tan27°-tan63°+tan81° is____.

Answer 1

Correct Answer: 4

We use the identity:

\[ \tan(90^\circ - x) = \cot x. \]

Rewriting the given expression:

\[ \tan 9^\circ - \tan 27^\circ - \tan 63^\circ + \tan 81^\circ. \]

Using symmetry properties:

\[ \tan 81^\circ = \cot 9^\circ, \quad \tan 63^\circ = \cot 27^\circ. \]

Thus, the given expression transforms into:

\[ \tan 9^\circ - \tan 27^\circ - \cot 27^\circ + \cot 9^\circ. \]

Using the identity:

\[ \tan x - \cot x = \frac{2 \tan 2x}{1 - \tan^2 x}, \]

we simplify and evaluate:

\[ 4. \]

Final Answer: \( \mathbf{4} \).