The Value of \( \tan75^\circ-\cot75^\circ \)
Question
Find the value of
\[ \tan75^\circ-\cot75^\circ \]
(a) \(2\sqrt3\)
(b) \(2+\sqrt3\)
(c) \(2-\sqrt3\)
(d) \(1\)
Solution
Use the identity
\[ \tan\theta-\cot\theta = \frac{\sin^2\theta-\cos^2\theta} {\sin\theta\cos\theta} \]
\[ = \frac{-\cos2\theta} {\frac12\sin2\theta} \]
\[ = -2\cot2\theta \]
Putting
\[ \theta=75^\circ \]
\[ \tan75^\circ-\cot75^\circ = -2\cot150^\circ \]
Since
\[ \cot150^\circ = -\sqrt3 \]
Therefore,
\[ -2(-\sqrt3) = 2\sqrt3 \]
Final Answer
\[ \boxed{2\sqrt3} \]
Hence, the correct option is (a) \(2\sqrt3\).