Rewrite 1-cos (135)/sin (135) using half-angled idenitity

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