Okay now a really really interesting question heh

Well first part is very easy to proof. Now second part.
Hence would imply using the previous part to solve the next part.
Now


 = -1)
 = -1)
[I'm gonna leave out my steps for solving for x cause it's trivial]
Solving for x yields

for

Now let's use the "otherwise" method.

 - \frac{\sqrt{2}}{2}sin(x) = \frac{\sqrt{2}}{2})
cos(x) - sin(\frac{\pi}{4})sin(x) = \frac{\sqrt{2}}{2})
 = \frac{\sqrt{2}}{2})
Solving for x yields

My question is why does one method give only 1 solution and another method gives 3? Which is the right method? When I type to TI-89 calc to just solve
 = cos(x) - 1)
with restricted domain I get all 3 solutions. However the answer for this question is just

I know why the "hence" method only gives 1 solution is because when you divide both sides by
 - 1)
and if

then it is undefined, but it still does not explain why you can not have 0 and

as solutions because if you just have
 = cos(x) - 1)
then it clearly satisfies.
Ideas anyone?