To solve the equation [tex]\(\sin \left(x+\frac{\pi}{4}\right)-\sin \left(x-\frac{\pi}{4}\right)=1\)[/tex], we will apply the sum and difference formulas for sine.
The sum and difference identities for sine are:
1. [tex]\(\sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b)\)[/tex]
2. [tex]\(\sin(a - b) = \sin(a)\cos(b) - \cos(a)\sin(b)\)[/tex]
Applying these identities to the given equation:
[tex]\[
\sin \left(x+\frac{\pi}{4}\right) = \sin(x)\cos\left(\frac{\pi}{4}\right) + \cos(x)\sin\left(\frac{\pi}{4}\right)
\][/tex]
[tex]\[
\sin \left(x-\frac{\pi}{4}\right) = \sin(x)\cos\left(\frac{\pi}{4}\right) - \cos(x)\sin\left(\frac{\pi}{4}\right)
\][/tex]
Now, subtract the second equation from the first:
[tex]\[
\sin \left(x+\frac{\pi}{4}\right)-\sin \left(x-\frac{\pi}{4}\right) = \left(\sin(x)\cos\left(\frac{\pi}{4}\right) + \cos(x)\sin\left(\frac{\pi}{4}\right)\right) - \left(\sin(x)\cos\left(\frac{\pi}{4}\right) - \cos(x)\sin\left(\frac{\pi}{4}\right)\right)
\][/tex]
Simplifying the right-hand side:
[tex]\[
\sin(x)\cos\left(\frac{\pi}{4}\right) + \cos(x)\sin\left(\frac{\pi}{4}\right) - \sin(x)\cos\left(\frac{\pi}{4}\right) + \cos(x)\sin\left(\frac{\pi}{4}\right)
\][/tex]
The terms involving [tex]\(\sin(x)\cos\left(\frac{\pi}{4}\right)\)[/tex] cancel each other out:
[tex]\[
\sin \left(x+\frac{\pi}{4}\right)-\sin \left(x-\frac{\pi}{4}\right) = 2 \cos(x)\sin\left(\frac{\pi}{4}\right)
\][/tex]
Next, we know that [tex]\(\cos\left(\frac{\pi}{4}\right) = \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\)[/tex], so the equation simplifies to:
[tex]\[
2 \cos\left(\frac{\pi}{4}\right) \cdot \sin(x) = 2 \cdot \frac{\sqrt{2}}{2} \cdot \sin(x) = \sqrt{2} \sin(x)
\][/tex]
Finally, we equate this to the right-hand side of the original equation:
[tex]\[
\sqrt{2} \sin(x) = 1
\][/tex]
Thus, after applying the sum and difference formulas for sine, the equation becomes:
[tex]\[
\sqrt{2} \sin(x) = 1
\][/tex]
