If `tantheta=(a)/(b)`,then find the value of `(costheta+sintheta)/(costheta-sintheta)`.

Answer

To solve the problem, we start with the given information and manipulate the expression step by step.

### Step 1: Write down the given information
We are given that:
\[ \tan \theta = \frac{a}{b} \]
We need to find the value of:
\[ \frac{\cos \theta + \sin \theta}{\cos \theta - \sin \theta} \]

### Step 2: Express \(\sin \theta\) and \(\cos \theta\) in terms of \(\tan \theta\)
Using the definition of tangent, we can express \(\sin \theta\) and \(\cos \theta\) in terms of \(a\) and \(b\):
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{a}{b} \]
This implies:
\[ \sin \theta = \frac{a}{\sqrt{a^2 + b^2}} \quad \text{and} \quad \cos \theta = \frac{b}{\sqrt{a^2 + b^2}} \]

### Step 3: Substitute \(\sin \theta\) and \(\cos \theta\) into the expression
Now we substitute these values into the expression:
\[ \frac{\cos \theta + \sin \theta}{\cos \theta - \sin \theta} = \frac{\frac{b}{\sqrt{a^2 + b^2}} + \frac{a}{\sqrt{a^2 + b^2}}}{\frac{b}{\sqrt{a^2 + b^2}} - \frac{a}{\sqrt{a^2 + b^2}}} \]

### Step 4: Simplify the expression
Since both the numerator and denominator have a common factor of \(\sqrt{a^2 + b^2}\), we can cancel it out:
\[ = \frac{b + a}{b - a} \]

### Step 5: Final result
Thus, the value of \(\frac{\cos \theta + \sin \theta}{\cos \theta - \sin \theta}\) is:
\[ \frac{a + b}{b - a} \]