If $\displaystyle \tan \theta = \frac{1}{\sqrt7}$ and $\theta$ is an acute angle, find $\displaystyle \frac{\text{cosec}^2 \theta - \sec^2 \theta}{\text{cosec}^2 \theta + \sec^2 \theta}$.

Using the identity cot squared theta equals cosec squared theta minus 1 and tan squared theta equals sec squared theta minus 1, the numerator becomes 1 minus tan squared theta and the denominator becomes 1 plus tan squared theta. Substituting tan theta equal to 1 divided by the square root of 7 gives tan squared theta equal to 1 divided by 7. The numerator is 1 minus 1 divided by 7, which is 6 divided by 7, and the denominator is 1 plus 1 divided by 7, which is 8 divided by 7. Dividing the two gives 6 divided by 8. The result is 3 divided by 4.

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