The value of tan 9°-tan27°-tan63°+tan81° is____.
Correct Answer: 4
Solution and Explanation
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: \( \mathbf{4} \).
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