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.