cos\left(2x\right)+sin\left(2x\right)=\frac{\sqrt{3}+1}{2}

To solve the equation \( \cos(2x) + \sin(2x) = \frac{\sqrt{3} + 1}{2} \), follow these steps:

### Step 1: Simplify the Equation
We know that \( \cos(2x) + \sin(2x) \) can be rewritten using the sum-to-product identity:
\[
\cos(2x) + \sin(2x) = \sqrt{2} \sin\left(2x + \frac{\pi}{4}\right)
\]
Thus, the equation becomes:
\[
\sqrt{2} \sin\left(2x + \frac{\pi}{4}\right) = \frac{\sqrt{3} + 1}{2}
\]

### Step 2: Isolate the Sine Function
Divide both sides by \( \sqrt{2} \) to isolate the sine function:
\[
\sin\left(2x + \frac{\pi}{4}\right) = \frac{\sqrt{3} + 1}{2\sqrt{2}}
\]

### Step 3: Simplify the Right Side
Simplify the right-hand side:
\[
\sin\left(2x + \frac{\pi}{4}\right) = \frac{\sqrt{6} + \sqrt{2}}{4}
\]

### Step 4: Solve the Trigonometric Equation
Now, solve for \( 2x + \frac{\pi}{4} \):
\[
2x + \frac{\pi}{4} = \arcsin\left(\frac{\sqrt{6} + \sqrt{2}}{4}\right)
\]
or
\[
2x + \frac{\pi}{4} = \pi - \arcsin\left(\frac{\sqrt{6} + \sqrt{2}}{4}\right)
\]

### Step 5: Solve for \( x \)
For the first case:
\[
2x = \arcsin\left(\frac{\sqrt{6} + \sqrt{2}}{4}\right) - \frac{\pi}{4}
\]
\[
x = \frac{1}{2}\left(\arcsin\left(\frac{\sqrt{6} + \sqrt{2}}{4}\right) - \frac{\pi}{4}\right)
\]

For the second case:
\[
2x = \pi - \arcsin\left(\frac{\sqrt{6} + \sqrt{2}}{4}\right) - \frac{\pi}{4}
\]
\[
x = \frac{1}{2}\left(\pi - \arcsin\left(\frac{\sqrt{6} + \sqrt{2}}{4}\right) - \frac{\pi}{4}\right)
\]

### Final Solution
The general solutions are:
\[
x = \frac{1}{2}\left(\arcsin\left(\frac{\sqrt{6} + \sqrt{2}}{4}\right) - \frac{\pi}{4}\right) + n\pi
\]
or
\[
x = \frac{1}{2}\left(\pi - \arcsin\left(\frac{\sqrt{6} + \sqrt{2}}{4}\right) - \frac{\pi}{4}\right) + n\pi
\]
where \( n \) is any integer.

Would you like more details on any step or have any questions?

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Here are 8 related questions to expand your understanding:
1. How do you derive the sum-to-product identity used in trigonometry?
2. What are the properties of the sine function that allow multiple solutions for \( x \)?
3. How can you verify the solution to a trigonometric equation?
4. What is the significance of the arcsin function in solving trigonometric equations?
5. How does the periodic nature of trigonometric functions affect the solution set?
6. Can you solve the equation using a different trigonometric identity?
7. What are the steps to graphically solve a trigonometric equation?
8. How do different trigonometric equations transform the unit circle?

**Tip:** Always double-check your trigonometric identities and transformations to avoid common mistakes when solving equations.

Learn how to solve the trigonometric equation cos(2x) + sin(2x) = (sqrt(3) + 1)/2 step-by-step. This problem involves using trigonometric identities and the arcsin function to find solutions. Suitable for advanced high school students and beyond.

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