$$ \tan\left(x+\frac14\cdot\pi\right)=\frac{\cos x+\sin x}{\cos x-\sin x} $$evaluate

We are given the equation:

\[
\tan\left(x + \frac{\pi}{4}\right) = \frac{\cos x + \sin x}{\cos x - \sin x}
\]

### Step 1: Simplify the left-hand side using the angle addition formula for tangent.
The angle addition formula for tangent is:

\[
\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \cdot \tan B}
\]

In this case, $ A = x $ and $ B = \frac{\pi}{4} $, and we know that:

\[
\tan\left(\frac{\pi}{4}\right) = 1
\]

So, applying the formula:

\[
\tan\left(x + \frac{\pi}{4}\right) = \frac{\tan x + 1}{1 - \tan x}
\]

Now the equation becomes:

\[
\frac{\tan x + 1}{1 - \tan x} = \frac{\cos x + \sin x}{\cos x - \sin x}
\]

### Step 2: Express the right-hand side in terms of tangent.
The right-hand side of the equation, $\frac{\cos x + \sin x}{\cos x - \sin x}$, can be simplified. Divide both the numerator and denominator by $\cos x$:

\[
\frac{\cos x + \sin x}{\cos x - \sin x} = \frac{1 + \tan x}{1 - \tan x}
\]

Thus, the equation now becomes:

\[
\frac{\tan x + 1}{1 - \tan x} = \frac{1 + \tan x}{1 - \tan x}
\]

### Step 3: Conclude the result.
Since both sides of the equation are equal, the equation holds for all $x$ where the tangent function is defined. Therefore, the equation is an identity, meaning it is true for all $x$ in the domain of $\tan x$, except where $ \tan x = 1 $, which would cause the denominator to be zero.

Would you like to explore any further details or have questions?

### Here are 5 related questions for further understanding:
1. How do you derive the tangent addition formula?
2. Can we generalize this process for other angle sums like $\tan(x + \theta)$?
3. What happens to the identity when $\tan x = 1$?
4. How would this process change for $\sin(x + \theta)$ or $\cos(x + \theta)$?
5. What is the geometric interpretation of tangent and how does it apply here?

### Tip:
When working with trigonometric identities, always simplify expressions by breaking them down into basic trigonometric functions like sine and cosine to see common patterns.

$$ \tan\left(x+\frac{\pi}{4}\right)=\frac{\cos x+\sin x}{\cos x-\sin x} $$ evaluate

Learn how to evaluate the trigonometric identity $ \tan(x + \frac{\pi}{4}) = \frac{\cos x + \sin x}{\cos x - \sin x} $ by simplifying the tangent addition formula. This solution explains key concepts such as trigonometric identities and angle addition formulas. Suitable for students in Grades 10-12.

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