The relation between `alpha` and `beta` of a transistor is

Explanation

The dc common-base current ratio or current gain `(alpha_(dc))` is defined as the ratio of the collector current `(I_(C))` to the emitter current `(I_(E)).`
`alpha_(dc)=(I_(C))/(I_(E))`
The dc common-emitter current ratio or current gain `(beta_(dc))` is defined as the ratio of the collector current to the base current `(I_(B)).`
`beta_(dc)=(I_(C))/(I_(B))`
Since the emitter current `I_(E)=I_(B)+I_(C)`,
`(I_(E))/(I_(C))=(I_(B))/(I_(C))+1`
`therefore (1)/(alpha_(dc))=(1)/(beta_(dc))+1`
Therefore, the common-base current gain in terms of the common-emitter current gain is `alpha_(dc)=(beta_(dc))/(1+beta_(dc))`
Or, the common-emitter gain in terms of the common-base current gain is `beta_(dc)=(alpha_(dc))/(1-alpha_(dc))`