The value of the expression $$\frac{1}{4} \left\{\left(a + \frac{1}{a}\right)^2 - \left(a - \frac{1}{a}\right)^2\right\}$$ is:
$$\frac{1}{4} \left\{\left(a + \frac{1}{a}\right)^2 - \left(a - \frac{1}{a}\right)^2\right\}$$
$$\frac{1}{4}\left\{a^2+\frac{1}{a^2}+2-\left(a^2+\frac{1}{a^2}-2\right)\right\}$$
$$\frac{1}{4}\left\{a^2+\frac{1}{a^2}+2-a^2-\frac{1}{a^2}+2\right\}$$
$$\frac{1}{4}\times4=1$$
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