Question
What is the value of $\frac{\tan^2\theta-\sec^2\theta}{\cot^2\theta-\text{cosec}^2\theta}?$
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Answer
$\frac{\tan^2\theta-\sec^2\theta}{\cot^2\theta-\text{cosec}^2\theta}=\frac{-(\sec^2\theta-\tan^2\theta)}{-(\text{cosec}^2\theta-\cot^2\theta)}$
$=\frac{-1}{-1}=1 \ \begin{cases}\because \sec^2\theta-\tan^2\theta= 1 \\ \text{cosec}^2\theta-\cot^2\theta=1 \end{cases}$
$=\frac{-1}{-1}=1 \ \begin{cases}\because \sec^2\theta-\tan^2\theta= 1 \\ \text{cosec}^2\theta-\cot^2\theta=1 \end{cases}$
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