mosfet - MOS Current Mirror

I was investigating a MOS current mirror on the photo however i could not understand how the equation (3) was obtained from equation (2). I put a question mark also there on the photo.

My second question is that the main advantage of connecting base and collector in a current mirror. What are the advantages of using current mirrors? If you share your idea i will be happy.

(There must be a connection between drain and gate of M1. I forgot to draw it.)

Answer

First you need a connection between drain and gate for you current mirror to work properly.

To answer your question, the step from (2) to (3) involves an approximation. The term $\lambda Vds$ is usually quite small, and therefore

$$\frac{1}{1+\lambda V_{ds1}} \approx 1 - \lambda V_{ds1}$$ in the next step (after multiplication) the the term $ \lambda^2 V_{ds2} V_{ds1} $ is neglected which results in the approximation $$(1 + \lambda V_{ds2}) (1 - \lambda V_{ds1}) \approx 1 + \lambda (V_{ds2} - V_{ds1})$$