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$$V_o = \frac{V_s \cdot R_2}{R_1 \cdot (1+R_2/R_L)+R_2}$$

I need to derive this equation given this circuit:

Schematic

This is what I have so far.

$$I=\dfrac{V_s}{R_1+\dfrac{R_2 \cdot R_L}{R_2+R_L}}$$

$$V_{R_L} = I \cdot R_L$$

Where am I going wrong?

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If R\$_L\$ wasn't present the voltage output would just be: -

\$V_{S}\times\dfrac{R_2}{R_1 + R_2}\$

So, adjust R2 mathematically to account for R\$_L\$ being in parallel with it.

Can you take it from here?

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Your equation for \$I\$ is correct. The next step is

$$V_o = I \cdot (R_2||R_L)$$

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$$ I = \dfrac{V_s}{R_1+\dfrac{R_2 \cdot R_L}{R_2+R_L}} $$ is correct.

\$ V_{R_L} = I \cdot R_L\$ is the wrong part.

$$ V_{R_L} = I \cdot R_{\text{eq}}; R_{\text{eq}} = R_L|| R_2 = \dfrac{R_LR_2}{R_L+R_2} $$

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