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