verify each identity

Question

tan(x/2)=tanx/secx+1

Answer 1

For this to be true, you must mean
tan(x/2) = tanx/(secx+1)

let Ø = x/2

then we are proving that
tanØ = tan 2Ø/(sec 2Ø + 1)

RS = (sin 2Ø/cos 2Ø)/(1/cos 2Ø + 1)
= (sin 2Ø /cos 2Ø)/( (1 + cos 2Ø)/cos 2Ø
= sin 2Ø/(1 + cos 2Ø)
= 2sinØcosØ/(1 + 2cos^2 Ø - 1)
= sinØ/cosØ
= tanØ or tan (x/2)
= LS