Which one of the following pairs is correctly matched?

Options

A. If `x=(1+sin60^@-cos60^@)/(1+sin60^@+cos60^@)` then `x=tan60^@`
B. if `x=(1+sin90^@-cos90^@)/(1+sin90^@-cos90^@)` then `x=tan30^@`
C. If `x=(2tan30^@)/(1-tan^2 30^@)` then `x=tan60^@`
D. If `x=(1-tan^2 30^@)/(1+tan^2 30^@)` then `x=cos60^@`

Answer

To solve the question of which pair is correctly matched, we will analyze each option step by step.

### Step 1: Analyze Option 1
The first option states:
\[ x = \frac{1 + \sin 60^\circ - \cos 60^\circ}{1 + \sin 60^\circ + \cos 60^\circ} \]
and claims:
\[ x = \tan 60^\circ \]

**Calculating \( x \):**
1. **Values of sine and cosine:**
- \( \sin 60^\circ = \frac{\sqrt{3}}{2} \)
- \( \cos 60^\circ = \frac{1}{2} \)

2. **Substituting these values into the equation:**
\[ x = \frac{1 + \frac{\sqrt{3}}{2} - \frac{1}{2}}{1 + \frac{\sqrt{3}}{2} + \frac{1}{2}} \]
Simplifying the numerator:
\[ = \frac{1 + \frac{\sqrt{3}}{2} - \frac{1}{2}}{1 + \frac{\sqrt{3}}{2} + \frac{1}{2}} = \frac{1 + \frac{\sqrt{3} - 1}{2}}{1 + \frac{\sqrt{3} + 1}{2}} = \frac{\frac{2 + \sqrt{3} - 1}{2}}{\frac{2 + \sqrt{3} + 1}{2}} = \frac{1 + \sqrt{3}}{3 + \sqrt{3}} \]

3. **Now we simplify:**
- Multiply numerator and denominator by the conjugate of the denominator:
\[ x = \frac{(1 + \sqrt{3})(3 - \sqrt{3})}{(3 + \sqrt{3})(3 - \sqrt{3})} = \frac{3 + \sqrt{3} - 3 - \sqrt{3}}{9 - 3} = \frac{0}{6} = 0 \]

4. **Calculating \( \tan 60^\circ \):**
\[ \tan 60^\circ = \sqrt{3} \]

**Conclusion for Option 1:**
Since \( x \neq \tan 60^\circ \), Option 1 is incorrect.

### Step 2: Analyze Option 2
The second option states:
\[ x = \frac{1 + \sin 90^\circ - \cos 90^\circ}{1 + \sin 90^\circ + \cos 90^\circ} \]
and claims:
\[ x = \tan 30^\circ \]

**Calculating \( x \):**
1. **Values of sine and cosine:**
- \( \sin 90^\circ = 1 \)
- \( \cos 90^\circ = 0 \)

2. **Substituting these values into the equation:**
\[ x = \frac{1 + 1 - 0}{1 + 1 + 0} = \frac{2}{2} = 1 \]

3. **Calculating \( \tan 30^\circ \):**
\[ \tan 30^\circ = \frac{1}{\sqrt{3}} \]

**Conclusion for Option 2:**
Since \( x \neq \tan 30^\circ \), Option 2 is incorrect.

### Step 3: Analyze Option 3
The third option states:
\[ x = \frac{2 \tan 30^\circ}{1 - \tan^2 30^\circ} \]
and claims:
\[ x = \tan 60^\circ \]

**Calculating \( x \):**
1. **Calculating \( \tan 30^\circ \):**
\[ \tan 30^\circ = \frac{1}{\sqrt{3}} \]

2. **Substituting this value into the equation:**
\[ x = \frac{2 \cdot \frac{1}{\sqrt{3}}}{1 - \left(\frac{1}{\sqrt{3}}\right)^2} = \frac{\frac{2}{\sqrt{3}}}{1 - \frac{1}{3}} = \frac{\frac{2}{\sqrt{3}}}{\frac{2}{3}} = \frac{2}{\sqrt{3}} \cdot \frac{3}{2} = \frac{3}{\sqrt{3}} = \sqrt{3} \]

3. **Calculating \( \tan 60^\circ \):**
\[ \tan 60^\circ = \sqrt{3} \]

**Conclusion for Option 3:**
Since \( x = \tan 60^\circ \), Option 3 is correct.

### Step 4: Analyze Option 4
Since we found Option 3 to be correct, we do not need to analyze Option 4.

### Final Answer:
The correct answer is **Option 3**.

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