Let's simplify the expression:

1 + sinα / 2 cosα + sinα

We can multiply both the numerator and denominator by cosα to simplify the expression:

= (cosα + sinα) / (2 cosα + sinα cosα)
= (cosα + sinα) / (2 cosα + sin²α)
= cosα + sinα / 2cosα + 1 - cos²α
= cosα + sinα / 2cosα + sin²α
= cosα + sinα / (cosα)^2 + (sinα)^2 + 2sinα cosα
= cosα + sinα / 1 + sin2α
= cosα + sinα / cos2α
= tan(α/2) + cosec(α)

Therefore, the simplified form of the expression is tan(α/2) + cosec(α).