We need to prove that :

sinx+ tanx / (1+ sec x ) = sinx

We will start from the left side.

(sinx + tanx)/ (1+ secx)

We know that tanx = sinx/cosx ans secx = 1/cosx.

Then, we will substitute with identities.

==> (sinx+ sinx/cosx) / (1+ 1/cosx)

==> [(sinx*cosx + sinx)/cosx] / [ ( cosx + 1)/cosx]

Reduce cosx.

==> (sinxcosx + sinx ) / (cosx + 1)

Now we will factor sinx from the numerator.

==> sinx*(cosx+1)/(cosx+1)

Now we will reduce cosx+1

==> sinx

**Then we proved that ( sinx+tanx)/(1+secx) = sinx.**