Question

How do you verify the identity `tan(x+45)=(1+tanx)/(1-tanx)`?

Answer 1

We know that `tantheta = sintheta/costheta`, so:

`sin(x + 45)/cos(x + 45) = (1 + sinx/cosx)/(1 - sinx/cosx)`

We use the sum formulae `sin(A + B) = sinAcosB + cosAsinB` and `cos(A + B) = cosAcosB - sinAsinB` to expand.

`(sinxcos(45) + cosxsin(45))/(cosxcos(45) - sinxsin(45)) = ((cosx + sinx)/cosx)/((cosx - sinx)/cosx)`

`(sinx(1/sqrt(2)) + cosx(1/sqrt(2)))/(cosx(1/sqrt(2)) - sinx(1/sqrt(2))) = (cosx + sinx)/(cosx) xx (cosx)/(cosx - sinx)`

`(1/sqrt2(sinx + cosx))/(1/sqrt(2)(cosx - sinx)) = (cosx + sinx)/(cosx - sinx)`

`(sinx + cosx)/(cosx- sinx) = (cosx+ sinx)/(cosx - sinx)`

`LHS = RHS`

Identity proved!

Hopefully this helps!

Answer 2

Prove trig expression

Explanation:

Use the trig identity:
`tan (a + b) = (tan a + tan b)/(1 - tan a.tan b`
We get:
`tan (x + 45) = (tan 45 + tan x)/(1 - tan x.tan 45)`
Trig table --> tan 45 = 1
There for:
`tan (x + 45) = (1 + tan x)/(1 - tan x)`