What is (\frac{1}{\cosec x-\cot x}) equal to?

Correct answer: (\cosec x+\cot x)

Direct answer: option C, \(\cosec x+\cot x\). Start with the meaning of reciprocal: \(1/(\cosec x-\cot x)\) is the number that multiplies \(\cosec x-\cot x\) to give 1. Using \(\cosec^2x-\cot^2x=1\), we factor the difference of squares: \((\cosec x-\cot x)(\cosec x+\cot x)=1\). Therefore the reciprocal is \(\cosec x+\cot x\), wherever the expressions are defined. Option A is \((\cosec x-\cot x)/(\cosec x-\cot x)=1\), so it is not generally the reciprocal. Option B similarly equals 1, because numerator and denominator are identical, so it is not the required expression. Option C gives \((\cosec x+\cot x)/1\), since the product with the original denominator is 1; hence it is correct. Option D is \((1/(\cosec x+\cot x))/(1/(\cosec x+\cot x))=1\), so it is also not the answer. Memory cue: to rationalise \(\cosec x-\cot x\), multiply by its conjugate \(\cosec x+\cot x\).