How to derive formula for emitter current flowing through Re?
This is not a full answer but much too long for a comment and should give you some pointers

We know \$ \beta = \dfrac{I_E}{I_B} \$, and \$ \alpha = \dfrac{I_C}{I_E} \$ so we can define \$ \alpha = \dfrac{\beta}{\beta+1} \$, we can ignore \$ \alpha \$.

Typically \$ \beta \ge 100 \$ and \$ V_{be} \approx 0.7 \text{V} \$.

We also know \$ I_E = \dfrac{V_E}{R_E} = \dfrac{V_B - V_{be}}{R_E}= \beta \cdot I_B\$ 

And Finally \$ I_B = \dfrac{V_{cc}-V_B}{R_1}-\dfrac{V_B}{R_2}\$

This gives us:

$$\dfrac{V_B - V_{be}}{R_E}= \beta \cdot \left( \dfrac{V_{cc}-V_B}{R_1}-\dfrac{V_B}{R_2} \right)  $$

The rest is just maths, solve for \$ V_B \$ and use \$ I_E = \dfrac{V_B - V_{be}}{R_E}\$ to get your answer.

I'll leave the rest as an exercise.



For most practical designs we assume \$ I_B = 0\$ which greatly simplifies this but I encourage you to try both to see how similar the results are.