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

Options

A. `cot (theta//2)`
B. `tan(theta//2)`
C. `sin theta`
D. `cos theta`

Answer

To solve the expression \(\frac{1 - \tan(90^\circ - \theta) + \sec(90^\circ - \theta)}{\tan(90^\circ - \theta) + \sec(90^\circ - \theta) + 1}\), we can follow these steps:

### Step 1: Use Trigonometric Identities
We know that:
- \(\tan(90^\circ - \theta) = \cot(\theta)\)
- \(\sec(90^\circ - \theta) = \csc(\theta)\)

Thus, we can rewrite the expression as:
\[ \frac{1 - \cot(\theta) + \csc(\theta)}{\cot(\theta) + \csc(\theta) + 1} \]

### Step 2: Substitute Values
For simplicity, let's assume \(\theta = 30^\circ\). Then:
- \(\cot(30^\circ) = \frac{1}{\tan(30^\circ)} = \frac{1}{\frac{1}{\sqrt{3}}} = \sqrt{3}\)
- \(\csc(30^\circ) = \frac{1}{\sin(30^\circ)} = \frac{1}{\frac{1}{2}} = 2\)

Substituting these values into the expression gives:
\[ \frac{1 - \sqrt{3} + 2}{\sqrt{3} + 2 + 1} \]

### Step 3: Simplify the Numerator and Denominator
Now, simplifying the numerator:
\[ 1 - \sqrt{3} + 2 = 3 - \sqrt{3} \]
And the denominator:
\[ \sqrt{3} + 2 + 1 = \sqrt{3} + 3 \]

So the expression now is:
\[ \frac{3 - \sqrt{3}}{\sqrt{3} + 3} \]

### Step 4: Rationalize the Expression
To simplify further, we can multiply the numerator and denominator by the conjugate of the denominator:
\[ \frac{(3 - \sqrt{3})(\sqrt{3} - 3)}{(\sqrt{3} + 3)(\sqrt{3} - 3)} \]

Calculating the denominator:
\[ (\sqrt{3})^2 - (3)^2 = 3 - 9 = -6 \]

Now for the numerator:
\[ (3 - \sqrt{3})(\sqrt{3} - 3) = 3\sqrt{3} - 9 - \sqrt{3}\cdot\sqrt{3} + 3\sqrt{3} = 6\sqrt{3} - 9 - 3 = 6\sqrt{3} - 12 \]

So the expression becomes:
\[ \frac{6\sqrt{3} - 12}{-6} = -(\sqrt{3} - 2) = 2 - \sqrt{3} \]

### Final Result
Thus, the value of the original expression is:
\[ 2 - \sqrt{3} \]