[tex]\bf a\cdot \cfrac{1}{2}\left(a+\cfrac{1}{a} \right)\impliedby \textit{let's distribute first}
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\cfrac{a}{2}\left(a+\cfrac{1}{a} \right)\implies \cfrac{a}{2}\cdot a+\cfrac{\underline{a}}{2}\cdot \cfrac{1}{\underline{a}}\implies \cfrac{a^2}{2}+\cfrac{1}{2}\implies \cfrac{a^2+1}{2}[/tex]
