Question

Prove that sec theta + tan theta = tan (45° + theta/2)

Answer 1

For ease of typing I will let Ø/2 = x, then Ø = 2x
So we are proving that
sec(2x) + tan(2x) = tan(45 + x)

RS = sin(45+x) / cos(45+x)
= (sin45cosx + cos45sinx)/(cos45cosx - sin45sinx)
= (1/√2 cosx + 1/√2 sinx)/(1/√2 cosx - 1/√2 sinx)
= (cox + sinx)/(cosx - sinx) , after factoring and canceling 1/√2
= (cox + sinx)/(cosx - sinx) * (cox + sinx)/(cox + sinx)
= (cos^2 x + 2sinxcosx + sin^2 x) / (cos^2 x - sin^2 x)
= (1 + 2sinxcosx)/cos 2x
= (1 + sin 2x)/cos 2x

LS = sec(2x) + tan(2x)
= 1/ cos 2x + sin 2x/cos 2x
= (1 + sin 2x)/cos 2x
= RS

Q.E.D.