p12p12p12p12Solutionp12 p12p12 p12 p12 1p12p12 1p12 sin

(p^(1/2)-p^(-1/2))(p^(1/2)+p^(-1/2))

[p^1/2 - p^-1/2][p^1/2 + p^-1/2]

= [p^1/2 - (1/p^1/2)][p^1/2 + (1/p^1/2)] since a^-m = 1/a^m

= [(p^1/2 p^1/2 - 1)/p^1/2][(p^1/2 p^1/2 + 1)/p^1/2]

= [(p^1/2^{+ 1/2} - 1)/p^1/2][(p^1/2^{+ 1/2} + 1)/p^1/2] since a^m * aⁿ = a^m+n

= [(p¹ - 1)(p¹ + 1)]/p¹

= (p -1)(p +1)/p

= (p² -1)/p since (a - b)(a + b) = a² - b²

= p²/p -1/p

= p^2-1 - p^-1 since 1/a^m = a^-m , a^m/aⁿ = a^m-n

= p - p^-1

Hence [p^1/2 - p^-1/2]/[p^1/2 + p^-1/2] = p - p^-1