Alternative Direct Substitution Approach

1. Start from the given equation:
   [
   costheta + sintheta = sqrt{2}costheta.
   ]
2. Rearranging, we get:
   [
   sintheta = (sqrt{2}-1)costheta.
   ]
3. Now consider what we need to prove:
   [
   costheta – sintheta = sqrt{2}sintheta.
   ]
4. Substitute (sintheta = (sqrt{2}-1)costheta) into the left-hand side:
   [
   costheta – sintheta = costheta – (sqrt{2}-1)costheta = costheta(2-sqrt{2}).
   ]
5. Substitute (sintheta = (sqrt{2}-1)costheta) into the right-hand side:
   [
   sqrt{2}sintheta = sqrt{2}(sqrt{2}-1)costheta = (2-sqrt{2})costheta.
   ]
6. Hence, both sides are equal:
   [
   costheta – sintheta = (2-sqrt{2})costheta
   quadtext{and}quad
   sqrt{2}sintheta = (2-sqrt{2})costheta.
   ]
   Since the left-hand side and the right-hand side simplify to the same expression, the proof is complete.