We are given the equation:

$\sin^2(\theta) + \cos^2(\theta) = 2\sin(\theta)\cos(\theta)$

Step 1: Use the Pythagorean identity

We know from the Pythagorean identity that:

$\sin^2(\theta) + \cos^2(\theta) = 1$

Substitute this into the equation:

$1 = 2\sin(\theta)\cos(\theta)$

Step 2: Simplify the equation

Now, divide both sides by 2:

$\frac{1}{2} = \sin(\theta)\cos(\theta)$

Step 3: Express the right-hand side using a trigonometric identity

Recall the double-angle identity for sine:

$\sin(2\theta) = 2\sin(\theta)\cos(\theta)$

So the equation becomes:

$\frac{1}{2} = \frac{1}{2} \sin(2\theta)$

Step 4: Solve for $\theta$

Now, multiply both sides by 2 to simplify:

$1 = \sin(2\theta)$

Thus, we have:

$\sin(2\theta) = 1$

Step 5: Solve for $\theta$

We know that $\sin(\alpha) = 1$ at $\alpha = \frac{\pi}{2} + 2k\pi$ for any integer $k$. Therefore, we have:

$2\theta = \frac{\pi}{2} + 2k\pi$

Step 6: Solve for $\theta$

Now, divide both sides by 2:

$\theta = \frac{\pi}{4} + k\pi$

Step 7: Find the values of $\theta$ in the interval $[0, 2\pi)$

We are asked to find $\theta$ such that $0 \leq \theta < 2\pi$. Let's consider possible values of $k$:

• For $k = 0$, $\theta = \frac{\pi}{4}$
• For $k = 1$, $\theta = \frac{\pi}{4} + \pi = \frac{5\pi}{4}$

Thus, the two values of $\theta$ in the interval $[0, 2\pi)$ are:

$\theta = \frac{\pi}{4} \quad \text{and} \quad \theta = \frac{5\pi}{4}$

Final Answer:

The angles $\theta$ that satisfy the given equation are:

$\boxed{\frac{\pi}{4} \text{ and } \frac{5\pi}{4}}$

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Would you like further explanation on any step? Here are some related questions to explore:

1. Derive the double-angle identity for sine and cosine.
2. How can you solve trigonometric equations with identities?
3. What are the other trigonometric identities that can simplify expressions like this?
4. How do you find the general solution for a sine equation like $\sin(x) = a$?
5. What is the significance of the unit circle in solving trigonometric equations?

Tip: Always try to simplify trigonometric equations using identities before solving for the variable to reduce complexity.