What is the value of `([tan(90-theta)+sec(90-theta)-1])/([tan(90-theta)-sec(90-theta)+1])`?

Options

A. `[1+costheta]/sintheta`
B. `[1+sintheta]/costheta`
C. `sintheta`
D. `costheta`

Answer

To solve the expression \(\frac{\tan(90^\circ - \theta) + \sec(90^\circ - \theta) - 1}{\tan(90^\circ - \theta) - \sec(90^\circ - \theta) + 1}\), we will use the trigonometric identities for \(\tan\) and \(\sec\).

### Step-by-step Solution:

1. **Use Trigonometric Identities**:
We know that:
\[ \tan(90^\circ - \theta) = \cot(\theta) \]
\[ \sec(90^\circ - \theta) = \csc(\theta) \]
Therefore, we can rewrite the expression as:
\[ \frac{\cot(\theta) + \csc(\theta) - 1}{\cot(\theta) - \csc(\theta) + 1} \]

2. **Substituting the Identities**:
Substitute the identities into the expression:
\[ \frac{\cot(\theta) + \csc(\theta) - 1}{\cot(\theta) - \csc(\theta) + 1} \]

3. **Combine Terms**:
Let's denote \(x = \cot(\theta)\) and \(y = \csc(\theta)\). The expression simplifies to:
\[ \frac{x + y - 1}{x - y + 1} \]

4. **Simplifying the Expression**:
Now we will simplify the expression:
- The numerator is \(x + y - 1\).
- The denominator is \(x - y + 1\).

5. **Factorizing**:
We can factor or rearrange the terms if necessary, but in this case, we will directly evaluate the expression:
\[ \frac{x + y - 1}{x - y + 1} \]

6. **Finding a Common Value**:
To find a common value, we can analyze specific values of \(\theta\). For example, if we take \(\theta = 45^\circ\):
- \(\cot(45^\circ) = 1\)
- \(\csc(45^\circ) = \sqrt{2}\)
Substituting these values gives:
\[ \frac{1 + \sqrt{2} - 1}{1 - \sqrt{2} + 1} = \frac{\sqrt{2}}{2 - \sqrt{2}} \]

7. **Final Evaluation**:
To simplify further, we can multiply the numerator and denominator by the conjugate of the denominator:
\[ \frac{\sqrt{2}(2 + \sqrt{2})}{(2 - \sqrt{2})(2 + \sqrt{2})} = \frac{2\sqrt{2} + 2}{4 - 2} = \frac{2\sqrt{2} + 2}{2} = \sqrt{2} + 1 \]

### Conclusion:
The value of the expression \(\frac{\tan(90^\circ - \theta) + \sec(90^\circ - \theta) - 1}{\tan(90^\circ - \theta) - \sec(90^\circ - \theta) + 1}\) simplifies to \(\sqrt{2} + 1\).