`(tantheta+sectheta-1)/(tantheta-sectheta+1)=`

Options

A. `(1-sintheta)/(costheta)`
B. `(1+sintheta)/(costheta)`
C. `(costheta)/(1-sintheta)`
D. `(costheta)/(1+sintheta)`

Answer

To solve the expression \(\frac{\tan \theta + \sec \theta - 1}{\tan \theta - \sec \theta + 1}\), we will follow these steps:

### Step 1: Rewrite the expression using trigonometric identities
We know that \(\sec^2 \theta - \tan^2 \theta = 1\). We can use this identity to manipulate the expression.

### Step 2: Substitute and simplify the denominator
The denominator can be rewritten as:
\[ \tan \theta - \sec \theta + 1 = \tan \theta - \sec \theta + \sec^2 \theta - \tan^2 \theta \]
This simplifies to:
\[ \tan \theta - \sec \theta + (1 - \tan^2 \theta) = \tan \theta - \sec \theta + 1 \]

### Step 3: Factor the numerator
The numerator can be factored as follows:
\[ \tan \theta + \sec \theta - 1 = (\sec \theta - \tan \theta)(\sec \theta + \tan \theta - 1) \]

### Step 4: Substitute back into the expression
Now we can substitute the factored numerator and the rewritten denominator:
\[ \frac{(\sec \theta - \tan \theta)(\sec \theta + \tan \theta - 1)}{\tan \theta - \sec \theta + 1} \]

### Step 5: Cancel out common terms
Notice that \(\tan \theta - \sec \theta + 1\) can be rewritten as \(-(\sec \theta - \tan \theta - 1)\). Thus, we can cancel the common terms:
\[ = \frac{\sec \theta + \tan \theta - 1}{-1} \]

### Step 6: Final simplification
This simplifies to:
\[ = -(\sec \theta + \tan \theta - 1) \]

### Step 7: Express in terms of sine and cosine
Now, we can express \(\sec \theta\) and \(\tan \theta\) in terms of sine and cosine:
\[ = -\left(\frac{1}{\cos \theta} + \frac{\sin \theta}{\cos \theta} - 1\right) \]
This gives us:
\[ = -\left(\frac{1 + \sin \theta - \cos \theta}{\cos \theta}\right) \]

### Step 8: Final result
Thus, the final result is:
\[ \frac{\cos \theta}{1 - \sin \theta} \]