If $a + \dfrac{1}{a} = p \text{ and } a - \dfrac{1}{a} = q$, find the relation between p and q.

We know that,

$\Rightarrow \Big(a + \dfrac{1}{a}\Big)^2 = a^2 + \dfrac{1}{a^2} + 2 \space …..(i)\\[1em] \Rightarrow \Big(a - \dfrac{1}{a}\Big)^2 = a^2 + \dfrac{1}{a^2} - 2 \space …..(ii)$

Subtracting eq. (ii) from (i) we get,

$\Rightarrow \Big(a + \dfrac{1}{a}\Big)^2 - \Big(a - \dfrac{1}{a}\Big)^2 = 4 \\[1em] \text{Substituting values we get}, \\[1em] \Rightarrow p^2 - q^2 = 4.$

Hence, p² - q² = 4.