\(2+\frac{sin\,\theta}{cot\,\theta - cosec\,\theta}\)
\(= 2+\cfrac{sin\,\theta}{\frac{cos\,\theta}{sin\,\theta} - \frac{1}{sin\,\theta}}\)
\(=2+\frac{sin^2\theta}{cos\,\theta -1}\)
\(=2 + \cfrac{4\,sin^2\frac{\theta}{2}\,cos^2\frac{\theta}{2}}{1 - 2\,sin^2\frac{\theta}{2}-1}\) \(\binom{\because\, sin\,\theta\, =\, 2\,sin\frac{\theta}{2}cos\frac{\theta}{2}}{\text{And}\,cos\,\theta\, =\, 1 - 2\,sin^2\frac{\theta}{2}}\)
\(= 2 - 2\,cos^2\frac{\theta}{2}\)
\(=2\left(1 - cos^2\frac{\theta}{2}\right)\)
\(=2\,sin^2\frac{\theta}{2}\) \(\left(\because sin^2\frac{\theta}{2} + cos^2\frac{\theta}{2} = 1\right)\)
Alternative : \(\rightarrow\)
\(2+\frac{sin\,\theta}{cot\,\theta - cosec\,\theta}\) \(=2+\frac{sin\,\theta}{cot\,\theta - cosec\,\theta}\times \frac{cot\,\theta + cosec\,\theta}{cot\,\theta + cosec\,\theta}\)
\(= 2 + \cfrac{sin\,\theta \left(\frac{cos\,\theta}{sin\,\theta} + \frac{1}{sin\,\theta}\right)}{cot^2\theta - cosec^2\theta}\)
= 2 - (cos θ + 1) (∵ cot2θ - cosec2θ = -1)
\(= 2 - \left(2\,cos^2 \frac{\theta}{2} - 1 + 1\right)\) \(\left(\because cos\,\theta = 2\,cos^2\frac{\theta}{2} - 1\right)\)
\(=2 - 2\,cos^2\frac{\theta}{2}\)
\(=2\left(1-cos^2\frac{\theta}{2}\right)\)
\(=2\,sin^2\frac{\theta}{2}\) \(\left(\because 1 - cos^2\frac{\theta}{2} = sin^2\frac{\theta}{2}\right)\)
