If $\tan \theta \, = \, \dfrac{1}{\sqrt2}$, find the value of $\dfrac{\text{cosec}^2 \theta \, - \, \sec^2 \theta}{\text{cosec}^2 \theta \, + \, \cot^2 \theta}$.

Using the identity sec squared theta equals 1 plus tan squared theta, we get sec squared theta equals 1 plus 1 over 2, which is 3 over 2, making cosec squared theta equal 1 plus 2 over 1, which is 3. Using the identity cosec squared theta equals 1 plus cot squared theta, cot squared theta equals 3 minus 1, which is 2. Substituting these values into the expression yields (3 minus 3 over 2) divided by (3 plus 2), simplifying to 3 over 2 divided by 5, which equals 3 over 10.

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