Answer
$R=\dfrac{R_1R_2}{R_1+R_2}$
Work Step by Step
The LCD on the right hand side is $R_1R_2$.
Make the expressions on the right hand side similar using their LCD.
$\dfrac{1}{R}=\dfrac{R_2}{R_1R_2}+\dfrac{R_1}{R_1R_2}$
Add numerators because denominators are same.
$\dfrac{1}{R}=\dfrac{R_2+R_1}{R_1R_2}$
Multiply both sides by $\frac{R\cdot R_1\cdot R_2}{R_1+R_2}$.
$\dfrac{RR_1R_2}{R_1+R_2}\cdot \dfrac{1}{R}=\dfrac{RR_1R_2}{R_1+R_2}\cdot \dfrac{R_2+R_1}{R_1R_2}$
Simplify by cancelling common factors.
$\require{cancel}
\begin{align*}
\dfrac{\cancel{R}R_1R_2}{R_1+R_2}\cdot \dfrac{1}{\cancel{R}}&=\dfrac{R\cancel{R_1R_2}}{\cancel{R_1+R_2}}\cdot \dfrac{\cancel{R_2+R_1}}{\cancel{R_1R_2}}\\
\\\dfrac{R_1R_2}{R_1+R_2}&=R
\end{align*}$
