\(tan\frac{x}{2}-cot\frac{x}{2}\)
\(= \frac{1}{2}.\frac{1}{cos^2\frac{x}{2}}+\frac{1}{2}.\frac{1}{sin^2\frac{x}{2}}\)
\(= \frac{1}{2}.\frac{cos^2\frac{x}{2}+sin^2\frac{x}{2}}{cos^2\frac{x}{2}.sin^2\frac{x}{2}}\)
\(= \frac{1}{2}.\frac{4}{[2.sin\frac{x}{2}.cos\frac{x}{2}]^2}\)
\(= \frac{2}{sin^2x}\)
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