To solve $\sin\left(\frac{\pi}{4}\right) + \cos\left(\frac{\pi}{4}\right)$:

1. $\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$
2. $\cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$

Now, adding these two values:

$\sin\left(\frac{\pi}{4}\right) + \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} = \frac{2\sqrt{2}}{2} = \sqrt{2}$

So, the result is $\sqrt{2}$.

Would you like a deeper explanation of these steps or any other questions?

Related Questions:

1. What are the values of $\sin\left(\frac{\pi}{3}\right) + \cos\left(\frac{\pi}{6}\right)$?
2. How do you prove the identity $\sin^2\theta + \cos^2\theta = 1$?
3. What is the value of $\sin\left(\frac{3\pi}{4}\right) + \cos\left(\frac{3\pi}{4}\right)$?
4. How do you calculate $\sin\left(\frac{5\pi}{4}\right) + \cos\left(\frac{5\pi}{4}\right)$?
5. Can you express $\sin\left(x + \frac{\pi}{4}\right)$ in terms of sine and cosine?

Tip: The angles $\frac{\pi}{4}, \frac{\pi}{6}, \frac{\pi}{3}$ are often encountered in trigonometry, so it's useful to memorize their sine and cosine values.