The value of `cosec^(2) 18^(@) - 1/(cot^(2) 72^(@)` is

Options

A. `(1)/(sqrt(3))`
B. `(sqrt(2))/(3)`
C. `(1)/(2)`
D. 1

Answer

To solve the problem, we need to find the value of \( \csc^2 18^\circ - \frac{1}{\cot^2 72^\circ} \).

### Step-by-step Solution:

1. **Understanding the Trigonometric Identities**:
We know that:
\[ \csc^2 \theta = 1 + \cot^2 \theta \]
Therefore, we can express \( \csc^2 18^\circ \) as:
\[ \csc^2 18^\circ = 1 + \cot^2 18^\circ \]

2. **Rewriting the Expression**:
We can rewrite the original expression:
\[ \csc^2 18^\circ - \frac{1}{\cot^2 72^\circ} \]
Since \( \cot 72^\circ = \tan(90^\circ - 72^\circ) = \tan 18^\circ \), we have:
\[ \cot^2 72^\circ = \tan^2 18^\circ \]
Thus, we can rewrite \( \frac{1}{\cot^2 72^\circ} \) as:
\[ \frac{1}{\cot^2 72^\circ} = \frac{1}{\tan^2 18^\circ} \]

3. **Substituting Back**:
Now substituting back into the expression:
\[ \csc^2 18^\circ - \frac{1}{\tan^2 18^\circ} = (1 + \cot^2 18^\circ) - \frac{1}{\tan^2 18^\circ} \]

4. **Using the Identity for Cotangent**:
We know that:
\[ \cot^2 18^\circ = \frac{1}{\tan^2 18^\circ} \]
Thus, we can simplify:
\[ \csc^2 18^\circ - \frac{1}{\tan^2 18^\circ} = 1 + \cot^2 18^\circ - \tan^2 18^\circ \]

5. **Final Simplification**:
Since \( \cot^2 \theta - \tan^2 \theta = 1 \) for any angle \( \theta \):
\[ \cot^2 18^\circ - \tan^2 18^\circ = 1 \]
Therefore, we have:
\[ 1 + 1 = 2 \]

6. **Conclusion**:
Thus, the final value of the expression \( \csc^2 18^\circ - \frac{1}{\cot^2 72^\circ} \) is:
\[ \boxed{2} \]