Question

tan 9$^\circ$ - tan 27$^\circ$ - tan 63$^\circ$ + tan 81$^\circ$ =

• A. 4
• B. 3
• C. 2
• D. 1

Hint:

For simplifications involving cotangent and tangent, look for complementary angle identities such as \(\tan (90^\circ - \theta) = \cot \theta\).

Answer 1

The Correct Option is A

We are given the expression: \[ \tan 9^\circ - \tan 27^\circ - \tan 63^\circ + \tan 81^\circ \] We can simplify this expression by recognizing the identities between the angles: \[ \tan 81^\circ = \cot 9^\circ \quad \text{and} \quad \tan 63^\circ = \cot 27^\circ \] Substituting these, the expression becomes: \[ \tan 9^\circ - \tan 27^\circ - \cot 27^\circ + \cot 9^\circ \] Now, since \(\tan \theta + \cot \theta = 2\), the expression simplifies to: \[ 2 (\tan 9^\circ - \tan 27^\circ) = 4 \] Thus, the final value of the expression is \(4\).