Consider the following
I.`(cot 30^@+1)/(cot30^@ -1) = 2 (cos300^@ +1)`
II. `2 sin 45^@ cos 45^@ - tan 45^@ cot 45^@ = 0`
Which of the above identities is/are correct ?

Options

A. Only I
B. Only II
C. Both I andII
D. Neither I nor II

Answer

To determine which of the given identities is correct, we will analyze each equation step by step.

### Step 1: Analyze the first identity
The first identity is:
\[ \frac{\cot 30^\circ + 1}{\cot 30^\circ - 1} = 2 (\cos 300^\circ + 1) \]

#### Step 1.1: Calculate \(\cot 30^\circ\)
We know that:
\[ \cot 30^\circ = \frac{1}{\tan 30^\circ} = \frac{1}{\frac{1}{\sqrt{3}}} = \sqrt{3} \]

#### Step 1.2: Substitute \(\cot 30^\circ\) into the equation
Substituting \(\cot 30^\circ\) into the left-hand side:
\[ \frac{\sqrt{3} + 1}{\sqrt{3} - 1} \]

#### Step 1.3: Simplify the left-hand side
To simplify, we rationalize the denominator:
\[ \frac{\sqrt{3} + 1}{\sqrt{3} - 1} \cdot \frac{\sqrt{3} + 1}{\sqrt{3} + 1} = \frac{(\sqrt{3} + 1)^2}{(\sqrt{3})^2 - (1)^2} = \frac{3 + 2\sqrt{3} + 1}{3 - 1} = \frac{4 + 2\sqrt{3}}{2} = 2 + \sqrt{3} \]

#### Step 1.4: Calculate \(\cos 300^\circ\)
Next, we find \(\cos 300^\circ\):
\[ \cos 300^\circ = \cos(360^\circ - 60^\circ) = \cos 60^\circ = \frac{1}{2} \]

#### Step 1.5: Substitute \(\cos 300^\circ\) into the right-hand side
Now substituting into the right-hand side:
\[ 2 \left(\frac{1}{2} + 1\right) = 2 \left(\frac{1}{2} + \frac{2}{2}\right) = 2 \cdot \frac{3}{2} = 3 \]

#### Step 1.6: Compare both sides
Now we compare both sides:
\[ 2 + \sqrt{3} \quad \text{and} \quad 3 \]
Since \(2 + \sqrt{3} \approx 3.732\) which is not equal to \(3\), the first identity is incorrect.

### Step 2: Analyze the second identity
The second identity is:
\[ 2 \sin 45^\circ \cos 45^\circ - \tan 45^\circ \cot 45^\circ = 0 \]

#### Step 2.1: Calculate \(\sin 45^\circ\) and \(\cos 45^\circ\)
We know:
\[ \sin 45^\circ = \cos 45^\circ = \frac{\sqrt{2}}{2} \]

#### Step 2.2: Substitute into the left-hand side
Substituting into the left-hand side:
\[ 2 \left(\frac{\sqrt{2}}{2}\right) \left(\frac{\sqrt{2}}{2}\right) - \tan 45^\circ \cot 45^\circ \]
\[ = 2 \cdot \frac{2}{4} - 1 \cdot 1 = 1 - 1 = 0 \]

#### Step 2.3: Conclusion for the second identity
Since the left-hand side equals the right-hand side, the second identity is correct.

### Final Conclusion
- The first identity is incorrect.
- The second identity is correct.

Thus, the answer is that only the second identity is correct.

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