Verify :
`"cosec "alpha=1+cot^(2)alpha" if "alpha=(pi)/(3)`.

Answer

To verify the identity \( \csc \alpha = 1 + \cot^2 \alpha \) for \( \alpha = \frac{\pi}{3} \), we will follow these steps:

### Step 1: Calculate \( \csc \left( \frac{\pi}{3} \right) \)

The cosecant function is the reciprocal of the sine function. Therefore,

\[ \csc \left( \frac{\pi}{3} \right) = \frac{1}{\sin \left( \frac{\pi}{3} \right)} \]

We know that \( \sin \left( \frac{\pi}{3} \right) = \frac{\sqrt{3}}{2} \). Thus,

\[ \csc \left( \frac{\pi}{3} \right) = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} \]

### Step 2: Calculate \( \cot \left( \frac{\pi}{3} \right) \)

The cotangent function is the reciprocal of the tangent function. Therefore,

\[ \cot \left( \frac{\pi}{3} \right) = \frac{1}{\tan \left( \frac{\pi}{3} \right)} \]

We know that \( \tan \left( \frac{\pi}{3} \right) = \sqrt{3} \). Thus,

\[ \cot \left( \frac{\pi}{3} \right) = \frac{1}{\sqrt{3}} \]

### Step 3: Calculate \( \cot^2 \left( \frac{\pi}{3} \right) \)

Now we will square the value of \( \cot \left( \frac{\pi}{3} \right) \):

\[ \cot^2 \left( \frac{\pi}{3} \right) = \left( \frac{1}{\sqrt{3}} \right)^2 = \frac{1}{3} \]

### Step 4: Calculate \( 1 + \cot^2 \left( \frac{\pi}{3} \right) \)

Now we will add 1 to \( \cot^2 \left( \frac{\pi}{3} \right) \):

\[ 1 + \cot^2 \left( \frac{\pi}{3} \right) = 1 + \frac{1}{3} = \frac{3}{3} + \frac{1}{3} = \frac{4}{3} \]

### Step 5: Compare \( \csc \left( \frac{\pi}{3} \right) \) and \( 1 + \cot^2 \left( \frac{\pi}{3} \right) \)

Now we will compare the two results:

\[ \csc \left( \frac{\pi}{3} \right) = \frac{2}{\sqrt{3}} \]

and

\[ 1 + \cot^2 \left( \frac{\pi}{3} \right) = \frac{4}{3} \]

To compare these, we can convert \( \frac{2}{\sqrt{3}} \) to have a common denominator:

\[ \frac{2}{\sqrt{3}} = \frac{2 \cdot \sqrt{3}}{3} = \frac{2\sqrt{3}}{3} \]

Now we need to check if \( \frac{2\sqrt{3}}{3} = \frac{4}{3} \).

### Step 6: Verify the equality

To verify the equality, we can square both sides:

\[ \left( \frac{2\sqrt{3}}{3} \right)^2 = \left( \frac{4}{3} \right)^2 \]

Calculating both sides:

\[ \frac{4 \cdot 3}{9} = \frac{16}{9} \]

Since both sides are equal, we conclude that:

\[ \csc \left( \frac{\pi}{3} \right) = 1 + \cot^2 \left( \frac{\pi}{3} \right) \]

Thus, the identity is verified.