To solve this problem step-by-step, let's start by simplifying the given equation and then solve for [tex]\(\theta\)[/tex].
### Given Equation:
[tex]\[
\cos \theta - \sin \theta = \sqrt{2} \sin \theta
\][/tex]
### Rearrange the Equation:
Move all the terms involving [tex]\(\sin \theta\)[/tex] to one side:
[tex]\[
\cos \theta = \sin \theta (\sqrt{2} + 1)
\][/tex]
### Solving for [tex]\(\theta\)[/tex]:
Now, let's find the angle [tex]\(\theta\)[/tex] that satisfies this equation.
Given:
[tex]\[
\cos \theta = \sin \theta (\sqrt{2} + 1)
\][/tex]
We seek the value of [tex]\(\theta\)[/tex] that satisfies this equation. The solutions for [tex]\(\theta\)[/tex] are:
[tex]\[
\theta = -2 \arctan(1 + \sqrt{2} + \sqrt{2} \sqrt{\sqrt{2} + 2}), \quad \theta = -2 \arctan(1 + \sqrt{2} - \sqrt{2} \sqrt{\sqrt{2} + 2})
\][/tex]
### Calculating [tex]\(\cos \theta + \sin \theta\)[/tex]:
Now, for the values of [tex]\(\theta\)[/tex] obtained, we want to find the value of [tex]\(\cos \theta + \sin \theta\)[/tex].
For each of these [tex]\(\theta\)[/tex]:
1. When [tex]\(\theta = -2 \arctan(1 + \sqrt{2} + \sqrt{2} \sqrt{\sqrt{2} + 2})\)[/tex], the value of [tex]\(\cos \theta + \sin \theta\)[/tex] is a value that we can denote as [tex]\(V_1\)[/tex].
2. When [tex]\(\theta = -2 \arctan(1 + \sqrt{2} - \sqrt{2} \sqrt{\sqrt{2} + 2})\)[/tex], the value of [tex]\(\cos \theta + \sin \theta\)[/tex] is a value that we can denote as [tex]\(V_2\)[/tex].
The specific value here for the simpler angle calculation would be:
[tex]\[
- \sin(2 \arctan(-\sqrt{2} \sqrt{\sqrt{2} + 2} + 1 + \sqrt{2})) + \cos(2 \arctan(-\sqrt{2} \sqrt{\sqrt{2} + 2} + 1 + \sqrt{2}))
\][/tex]
Thus, the value of [tex]\(\cos \theta + \sin \theta\)[/tex] for the simplified solution [tex]\(\theta\)[/tex] which is:
[tex]\[
- \sin(2 \arctan(-\sqrt{2} \sqrt{\sqrt{2} + 2} + 1 + \sqrt{2})) + \cos(2 \arctan(-\sqrt{2} \sqrt{\sqrt{2} + 2} + 1 + \sqrt{2}))
\][/tex]
This value represents the result of [tex]\(\cos(\theta) + \sin(\theta)\)[/tex] based on one of the solutions obtained for [tex]\(\theta\)[/tex].
