`(tan60^(@)+sec60^(@)-1)/(tan60^(@)-sec60^(@)+1)`A)`sec30^(@)+2cos60^(@)` B)`csc30^(@)+2sin60^(@)` C)`cot30^(@)+2tan60^(@)` D)none of these

Options

A. `sec30^(@)+2cos60^(@)`
B. `csc30^(@)+2sin60^(@)`
C. `cot30^(@)+2tan60^(@)`
D. none of these

Answer

To solve the expression \((\tan 60^\circ + \sec 60^\circ - 1) / (\tan 60^\circ - \sec 60^\circ + 1)\), we will follow these steps:

### Step 1: Substitute the values of \(\tan 60^\circ\) and \(\sec 60^\circ\)
We know:
- \(\tan 60^\circ = \sqrt{3}\)
- \(\sec 60^\circ = 2\)

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

### Step 2: Simplify the numerator and the denominator
Now, simplify the numerator and the denominator:
- Numerator: \(\sqrt{3} + 2 - 1 = \sqrt{3} + 1\)
- Denominator: \(\sqrt{3} - 2 + 1 = \sqrt{3} - 1\)

So the expression now looks like:
\[ \frac{\sqrt{3} + 1}{\sqrt{3} - 1} \]

### Step 3: Rationalize the denominator
To rationalize the denominator, we multiply the numerator and the denominator by the conjugate of the denominator, which is \(\sqrt{3} + 1\):
\[ \frac{(\sqrt{3} + 1)(\sqrt{3} + 1)}{(\sqrt{3} - 1)(\sqrt{3} + 1)} \]

### Step 4: Expand the numerator and the denominator
Now, we will expand both the numerator and the denominator:
- Numerator: \((\sqrt{3} + 1)(\sqrt{3} + 1) = 3 + 2\sqrt{3} + 1 = 4 + 2\sqrt{3}\)
- Denominator: \((\sqrt{3})^2 - (1)^2 = 3 - 1 = 2\)

So now we have:
\[ \frac{4 + 2\sqrt{3}}{2} \]

### Step 5: Simplify the fraction
Now, we can simplify this fraction:
\[ \frac{4}{2} + \frac{2\sqrt{3}}{2} = 2 + \sqrt{3} \]

### Step 6: Compare with the options
Now we need to check which option matches \(2 + \sqrt{3}\):
- Option A: \(\sec 30^\circ + 2\cos 60^\circ = \frac{2}{\sqrt{3}} + 2 \cdot \frac{1}{2} = \frac{2}{\sqrt{3}} + 1\) (not equal)
- Option B: \(\csc 30^\circ + 2\sin 60^\circ = 2 + 2 \cdot \frac{\sqrt{3}}{2} = 2 + \sqrt{3}\) (this matches)
- Option C: \(\cot 30^\circ + 2\tan 60^\circ = \frac{1}{\sqrt{3}} + 2\sqrt{3}\) (not equal)

Thus, the correct answer is **Option B**.