To prove these identities, we'll use trigonometric transformations and identities.

### a) Prove \( \sin x + \cos x = \sqrt{2} \sin \left( x + \frac{\pi}{4} \right) \).

#### Proof:

We use the angle addition formula for sine:
\[
\sin \left( x + \frac{\pi}{4} \right) = \sin x \cos \frac{\pi}{4} + \cos x \sin \frac{\pi}{4}
\]

Since \(\cos \frac{\pi}{4} = \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}\), we can rewrite it as:
\[
\sin \left( x + \frac{\pi}{4} \right) = \sin x \cdot \frac{\sqrt{2}}{2} + \cos x \cdot \frac{\sqrt{2}}{2}
= \frac{\sqrt{2}}{2}(\sin x + \cos x)
\]

Multiplying both sides by \(\sqrt{2}\), we get:
\[
\sqrt{2} \sin \left( x + \frac{\pi}{4} \right) = \sin x + \cos x
\]

Therefore, the identity is proven:
\[
\sin x + \cos x = \sqrt{2} \sin \left( x + \frac{\pi}{4} \right)
\]

---

### b) Prove \( \sin \left( x + \frac{\pi}{4} \right) - \sin \left( x - \frac{\pi}{4} \right) = \sqrt{2} \cos x \).

#### Proof:

Using the sine angle addition and subtraction formulas, we have:
\[
\sin \left( x + \frac{\pi}{4} \right) = \sin x \cos \frac{\pi}{4} + \cos x \sin \frac{\pi}{4}
\]
\[
\sin \left( x - \frac{\pi}{4} \right) = \sin x \cos \frac{\pi}{4} - \cos x \sin \frac{\pi}{4}
\]

Subtract the two equations:
\[
\sin \left( x + \frac{\pi}{4} \right) - \sin \left( x - \frac{\pi}{4} \right)
= \left( \sin x \cos \frac{\pi}{4} + \cos x \sin \frac{\pi}{4} \right) - \left( \sin x \cos \frac{\pi}{4} - \cos x \sin \frac{\pi}{4} \right)
\]

Simplify:
\[
= 2\cos x \sin \frac{\pi}{4}
= 2\cos x \cdot \frac{\sqrt{2}}{2}
= \sqrt{2} \cos x
\]

Thus, the identity is proven:
\[
\sin \left( x + \frac{\pi}{4} \right) - \sin \left( x - \frac{\pi}{4} \right) = \sqrt{2} \cos x
\]