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Moving coil galvanometer
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A galvanometer of resistance G is shunted by a resistance S ohm. To keep the main current in the circuit unchanged, the resistance to be put in series with the galvanometer isBefore shunting, the resistance offered by the galvanometer to the circuit is G.
After the shunt S is connected in parallel, the resistance of the combination is $\frac{GS}{G + S}$, which is less than G.
For the main current to remain unchanged, the total resistance must again be G. Let R be the resistance added in series with the shunted galvanometer.
$G = \frac{GS}{G + S} + R$
$R = G - \frac{GS}{G + S} = \frac{G^2 + GS - GS}{G + S}$
$R = \frac{G^2}{S + G}$