The two half angle identities we'll use are:
(1-cos2x)/2 =(sinx)^2 and sin2x = 2sinxcosx
first we'll deal with the numerator:
1-cos(135) kind of looks like 1-cos2x, so we'll use that identity after a little rearranging:
(1-cos2x)/2 =(sinx)^2
multiply both sides by 2 and we get:
(1-cos2x)= 2(sinx)^2
Now back to our numerator:
if 135 =2x, then x = 67.5
1-cos(135) = 2(sin67.5)^2
So we've rewritten our numerator as 2(sin67.5)^2
Now for our denominator we'll use the half angle identity:
sin2x = 2sinxcosx
So our denominator becomes:
sin(135) =2sin(67.5)cos(67.5)
Now put it all together...
(2(sin67.5)^2)/2sin(67.5)cos(67.5)
The 2 on top and bottom cancel and the (sin67.5)^2 cancels the sin67.5 on the bottom so you're left with
sin67.5/cos67.5
Which simplifes to
tan67.5
