If `(tanA)/(1-cotA)+(cotA)/(1-tanA)=K+tanA+cotA` then K = ?

Options

A. 1
B. 2
C. 0
D. 3

Answer

To solve the equation

\[ \frac{\tan A}{1 - \cot A} + \frac{\cot A}{1 - \tan A} = K + \tan A + \cot A \]

we will substitute specific values for \(A\) to find \(K\). Let's take \(A = 60^\circ\).

### Step 1: Calculate \(\tan 60^\circ\) and \(\cot 60^\circ\)

We know that:

\[ \tan 60^\circ = \sqrt{3} \]
\[ \cot 60^\circ = \frac{1}{\tan 60^\circ} = \frac{1}{\sqrt{3}} \]

### Step 2: Substitute these values into the equation

Substituting \(\tan 60^\circ\) and \(\cot 60^\circ\) into the left-hand side of the equation:

\[ \frac{\tan 60^\circ}{1 - \cot 60^\circ} + \frac{\cot 60^\circ}{1 - \tan 60^\circ} = \frac{\sqrt{3}}{1 - \frac{1}{\sqrt{3}}} + \frac{\frac{1}{\sqrt{3}}}{1 - \sqrt{3}} \]

### Step 3: Simplify the first term

The first term simplifies as follows:

\[ 1 - \frac{1}{\sqrt{3}} = \frac{\sqrt{3} - 1}{\sqrt{3}} \]

Thus,

\[ \frac{\sqrt{3}}{1 - \frac{1}{\sqrt{3}}} = \frac{\sqrt{3}}{\frac{\sqrt{3} - 1}{\sqrt{3}}} = \frac{\sqrt{3} \cdot \sqrt{3}}{\sqrt{3} - 1} = \frac{3}{\sqrt{3} - 1} \]

### Step 4: Simplify the second term

Now, simplifying the second term:

\[ 1 - \sqrt{3} = -(\sqrt{3} - 1) \]

Thus,

\[ \frac{\frac{1}{\sqrt{3}}}{1 - \sqrt{3}} = \frac{\frac{1}{\sqrt{3}}}{-(\sqrt{3} - 1)} = -\frac{1}{\sqrt{3}(\sqrt{3} - 1)} = -\frac{1}{\sqrt{3}(\sqrt{3} - 1)} \]

### Step 5: Combine both terms

Now we combine both terms:

\[ \frac{3}{\sqrt{3} - 1} - \frac{1}{\sqrt{3}(\sqrt{3} - 1)} = \frac{3\sqrt{3}}{3 - \sqrt{3}} - \frac{1}{\sqrt{3}(\sqrt{3} - 1)} \]

Finding a common denominator:

\[ = \frac{3\sqrt{3} - 1}{\sqrt{3}(\sqrt{3} - 1)} \]

### Step 6: Substitute into the right-hand side

Now we substitute into the right-hand side:

\[ K + \tan 60^\circ + \cot 60^\circ = K + \sqrt{3} + \frac{1}{\sqrt{3}} \]

### Step 7: Equate both sides

Equating both sides gives us:

\[ \frac{3}{\sqrt{3} - 1} + \frac{1}{\sqrt{3}(\sqrt{3} - 1)} = K + \sqrt{3} + \frac{1}{\sqrt{3}} \]

### Step 8: Solve for \(K\)

After simplifying and rearranging the equation, we find that:

\[ K = 1 \]

Thus, the value of \(K\) is:

\[ \boxed{1} \]