Find the value of ` : sqrt((1- sin ^(2) 60 ^(@))/(1- cos ^(2) 60^(@)))`

Answer

To find the value of

\[ \sqrt{\frac{1 - \sin^2 60^\circ}{1 - \cos^2 60^\circ}} \]

we will follow these steps:

### Step 1: Use the Pythagorean Identity
We know that

\[ \sin^2 \theta + \cos^2 \theta = 1 \]

From this identity, we can express \(1 - \sin^2 60^\circ\) and \(1 - \cos^2 60^\circ\).

### Step 2: Calculate \(1 - \sin^2 60^\circ\)
Using the identity, we have:

\[ 1 - \sin^2 60^\circ = \cos^2 60^\circ \]

### Step 3: Calculate \(1 - \cos^2 60^\circ\)
Similarly, we can express:

\[ 1 - \cos^2 60^\circ = \sin^2 60^\circ \]

### Step 4: Substitute these values into the expression
Now substituting these values into the original expression gives us:

\[ \sqrt{\frac{\cos^2 60^\circ}{\sin^2 60^\circ}} \]

### Step 5: Simplify the expression
This simplifies to:

\[ \sqrt{\left(\frac{\cos 60^\circ}{\sin 60^\circ}\right)^2} \]

### Step 6: Remove the square root
The square root and the square cancel each other out, so we have:

\[ \frac{\cos 60^\circ}{\sin 60^\circ} \]

### Step 7: Recognize the trigonometric function
This expression is equal to:

\[ \cot 60^\circ \]

### Step 8: Calculate \(\cot 60^\circ\)
We know that:

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

### Final Answer
Thus, the value of the original expression is:

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

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