cos 
2
 (x)−sin 
2
 (x)+1+cos(x)= 2sin(x)cos(x)+sin(x)/tan(x)
​


Let's solve the given trigonometric equation:

\[
\cos^2(x) - \sin^2(x) + 1 + \cos(x) = 2\sin(x)\cos(x) + \frac{\sin(x)}{\tan(x)}
\]

### Step 1: Simplify the equation

First, recognize that \(\frac{\sin(x)}{\tan(x)}\) simplifies as follows:
\[
\frac{\sin(x)}{\tan(x)} = \frac{\sin(x)}{\frac{\sin(x)}{\cos(x)}} = \cos(x)
\]
So, the equation becomes:
\[
\cos^2(x) - \sin^2(x) + 1 + \cos(x) = 2\sin(x)\cos(x) + \cos(x)
\]

### Step 2: Cancel out similar terms

Subtract \(\cos(x)\) from both sides:
\[
\cos^2(x) - \sin^2(x) + 1 = 2\sin(x)\cos(x)
\]

### Step 3: Apply trigonometric identities

Recall the identity for the difference of squares:
\[
\cos^2(x) - \sin^2(x) = \cos(2x)
\]
Thus, the equation simplifies to:
\[
\cos(2x) + 1 = 2\sin(x)\cos(x)
\]

### Step 4: Use another identity for the right side

Notice that \(2\sin(x)\cos(x)\) is the double-angle identity for sine:
\[
2\sin(x)\cos(x) = \sin(2x)
\]
Substituting this in gives:
\[
\cos(2x) + 1 = \sin(2x)
\]

### Step 5: Rearrange the equation

Move all terms to one side:
\[
\cos(2x) - \sin(2x) + 1 = 0
\]

### Step 6: Solve the equation

This equation is not straightforward to solve algebraically, but it can be approached by substituting specific values for \(x\), or by using graphical or numerical methods. However, by checking specific values or using an identity, we can potentially find solutions:

1. For \(x = 0\), \(\cos(2(0)) - \sin(2(0)) + 1 = 1 - 0 + 1 = 2 \neq 0\).
2. For \(x = \frac{\pi}{4}\), \( \cos\left(\frac{\pi}{2}\right) - \sin\left(\frac{\pi}{2}\right) + 1 = 0 - 1 + 1 = 0\).

Thus, \(x = \frac{\pi}{4}\) is a solution. There may be other solutions that can be found using more advanced methods or tools like graphing.

Would you like a more detailed explanation or further assistance?

Here are some related questions you might find useful:
1. What are the general solutions to the equation \( \cos(2x) - \sin(2x) + 1 = 0 \)?
2. How do trigonometric identities help in simplifying equations?
3. What are the properties of sine and cosine functions?
4. How can numerical methods be used to solve trigonometric equations?
5. What are the applications of double-angle identities in trigonometry?

**Tip:** When dealing with complex trigonometric equations, it’s often helpful to first simplify using known identities before attempting to solve.

Learn how to solve the trigonometric equation cos^2(x) - sin^2(x) + 1 + cos(x) = 2sin(x)cos(x) + sin(x)/tan(x) step-by-step with detailed explanations and examples. This problem involves trigonometric identities and equations, suitable for advanced high school students and beyond.

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