https://file.aiquickdraw.com/mathbot/file/1725458738035pifp4d1r.jpg

The equation provided in the image is:

\[
\sqrt{1 + \sin 2x} = \sin x + \cos x
\]

Let's solve this equation step by step.

### Step 1: Simplify the equation
We can start by squaring both sides to eliminate the square root:

\[
1 + \sin 2x = (\sin x + \cos x)^2
\]

### Step 2: Expand the right-hand side
Now, expand the square on the right-hand side:

\[
(\sin x + \cos x)^2 = \sin^2 x + 2\sin x \cos x + \cos^2 x
\]

We know that \(\sin^2 x + \cos^2 x = 1\), so:

\[
\sin^2 x + 2\sin x \cos x + \cos^2 x = 1 + 2\sin x \cos x
\]

Thus, the equation simplifies to:

\[
1 + \sin 2x = 1 + 2\sin x \cos x
\]

### Step 3: Simplify further
Since \(\sin 2x = 2\sin x \cos x\), we have:

\[
1 + \sin 2x = 1 + \sin 2x
\]

### Conclusion
The equation simplifies to \(1 + \sin 2x = 1 + \sin 2x\), which is an identity. This means the equation is true for all values of \(x\). Therefore, any \(x\) is a solution.

### Final Answer:
The given equation holds true for all real numbers \(x\).

Would you like more details on any of the steps or have any questions? Here are some related questions you might consider:

1. How would the solution change if the equation was \(\sqrt{1 + \sin 2x} = \cos x - \sin x\)?
2. What are the trigonometric identities used in simplifying \(\sin^2 x + \cos^2 x\)?
3. Can the equation \(\sqrt{1 + \sin 2x} = \sin x + \cos x\) have any extraneous solutions after squaring both sides?
4. How would you solve the equation if the square root was replaced with a different function, such as a cubic root?
5. How can the identity \(\sin 2x = 2\sin x \cos x\) be derived from basic trigonometric principles?

**Tip:** When solving trigonometric equations, always consider the possibility of extraneous solutions, especially when squaring both sides.

Solve the equation: \(\sqrt{1 + \sin 2x} = \sin x + \cos x\)

Explore how to solve the trigonometric equation \(\sqrt{1 + \sin 2x} = \sin x + \cos x\) using key trigonometric identities like the Pythagorean Identity. This problem is ideal for students in Grades 10-12 who are studying advanced trigonometry.

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