Take \(0^\circ<\theta<90^\circ\). Which one of these single terms is identical to the difference \(\displaystyle\frac{\sin\theta}{1-\cos\theta}-\frac{\sin\theta}{1+\cos\theta}\)?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(2\cot\theta\)
Take the common denominator \((1-\cos\theta)(1+\cos\theta)=1-\cos^2\theta\), which by \(\sin^2\theta+\cos^2\theta=1\) is \(\sin^2\theta\). In the given range \(\cos\theta\ne 1\) and \(\sin\theta\ne 0\), so nothing is divided by zero.
Numerator: \(\sin\theta(1+\cos\theta)-\sin\theta(1-\cos\theta)=2\sin\theta\cos\theta\)
Denominator: \(1-\cos^2\theta=\sin^2\theta\)
Expression \(=\displaystyle\frac{2\sin\theta\cos\theta}{\sin^2\theta}=\frac{2\cos\theta}{\sin\theta}=2\cot\theta\)