Question

How do you prove `Tan^2(x/2+Pi/4)=(1+sinx)/(1-sinx)`?

Answer 1

Proof below (it's a long one)

Explanation:

Ill work this backwards (but writing doing it forward would work as well):
`(1+sinx)/(1-sinx)=(1+sinx)/(1-sinx)*(1+sinx)/(1+sinx)`
`=(1+sinx)^2/(1-sin^2x)`
`=(1+sinx)^2/cos^2x`
`=((1+sinx)/cosx)^2`
Then substitute in `t` formula (Explanation below)
`=((1+(2t)/(1+t^2))/((1-t^2)/(1+t^2)))^2`
`=(((1+t^2+2t)/(1+t^2))/((1-t^2)/(1+t^2)))^2`
`=((1+t^2+2t)/(1-t^2))^2`
`=((1+2t+t^2)/(1-t^2))^2`
`=((1+t)^2/(1-t^2))^2`
`=((1+t)^2/((1-t)(1+t)))^2`
`=((1+t)/(1-t))^2`
`=((1+tan(x/2))/(1-tan(x/2)))^2`
`=((tan(pi/4)+tan(x/2))/(1-tan(x/2)tan(pi/4)))^2` Note that: (`tan(pi/4)=1)`
`=(tan(x/2+pi/4))^2`
`=tan^2(x/2+pi/4)`

T FORMULAE FOR THIS EQUATION:
`sinx=(2t)/(1-t^2)`, `cosx=(1-t^2)/(1+t^2)`, where `t=tan(x/2)`