Simplify: \(\left(1+\frac{q^2-p^2-r^2}{2pr}\right)\times\left(\frac{1}{q+p-r}+\frac{1}{r+q-p}\right)\)
\(\frac{q}{pr}\)
First bracket: \(1+\frac{q^2-p^2-r^2}{2pr} = \frac{q^2-(p^2-2pr+r^2)}{2pr} = \frac{q^2-(p-r)^2}{2pr}\).
Using \(a^2-b^2=(a-b)(a+b)\): \(\frac{(q-p+r)(q+p-r)}{2pr}\).
Second bracket: \(\frac{1}{q+p-r}+\frac{1}{r+q-p} = \frac{(r+q-p)+(q+p-r)}{(q+p-r)(r+q-p)} = \frac{2q}{(q+p-r)(r+q-p)}\).
Product: \(\frac{(q-p+r)(q+p-r)}{2pr}\times\frac{2q}{(q+p-r)(q-p+r)} = \frac{q}{pr}\).
Hence, the expression simplifies to \(\frac{q}{pr}\).
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