If \cot \theta = \sqrt{7}, then what is \frac{\operatorname{cosec}^2 \theta - \sec^2 \theta}{\operatorname{cosec}^2 \theta + \sec^2 \theta} equal to?

Explanation

Dividing the numerator and denominator by \sec^2 \theta transforms the expression to \frac{\cot^2 \theta - 1}{\cot^2 \theta + 1}. Substituting \cot^2 \theta = 7 gives \frac{7 - 1}{7 + 1} = \frac{6}{8} = \frac{3}{4}.

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