Find the value of :
`cosec^(2)60^(@)-tan^(2)30^(@)`

Answer

To solve the problem, we need to find the value of \( \csc^2 60^\circ - \tan^2 30^\circ \).

### Step-by-step Solution:

1. **Identify the trigonometric identities**:
\[ \csc^2 \theta = \frac{1}{\sin^2 \theta} \]
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \]
Therefore,
\[ \tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta} \]

2. **Substitute the angles into the identities**:
\[ \csc^2 60^\circ = \frac{1}{\sin^2 60^\circ} \]
\[ \tan^2 30^\circ = \frac{\sin^2 30^\circ}{\cos^2 30^\circ} \]

3. **Find the values of \( \sin 60^\circ \), \( \sin 30^\circ \), and \( \cos 30^\circ \)**:
- \( \sin 60^\circ = \frac{\sqrt{3}}{2} \)
- \( \sin 30^\circ = \frac{1}{2} \)
- \( \cos 30^\circ = \frac{\sqrt{3}}{2} \)

4. **Calculate \( \csc^2 60^\circ \)**:
\[ \csc^2 60^\circ = \frac{1}{\left(\frac{\sqrt{3}}{2}\right)^2} = \frac{1}{\frac{3}{4}} = \frac{4}{3} \]

5. **Calculate \( \tan^2 30^\circ \)**:
\[ \tan^2 30^\circ = \frac{\left(\frac{1}{2}\right)^2}{\left(\frac{\sqrt{3}}{2}\right)^2} = \frac{\frac{1}{4}}{\frac{3}{4}} = \frac{1}{3} \]

6. **Substitute the values back into the expression**:
\[ \csc^2 60^\circ - \tan^2 30^\circ = \frac{4}{3} - \frac{1}{3} = \frac{4 - 1}{3} = \frac{3}{3} = 1 \]

### Final Answer:
\[ \csc^2 60^\circ - \tan^2 30^\circ = 1 \]