If `cosec39^(@)=x`, then the value of
`(1)/(cosec^(2)51^(@))+sin^(2)39^(@)+tan^(2)51^(@)-(1)/(sin^(2)51^(@)sec^(2)39^(@))` is.

Options

A. `sqrt(x^(2)-1)`
B. `sqrt(1-x^(2))`
C. `x^(2)-1`
D. `1-x^(2)`

Answer

To solve the problem, we start with the given equation:

If \( \csc 39^\circ = x \), we need to find the value of:

\[ \frac{1}{\csc^2 51^\circ} + \sin^2 39^\circ + \tan^2 51^\circ - \frac{1}{\sin^2 51^\circ \sec^2 39^\circ} \]

### Step 1: Rewrite the terms using trigonometric identities

First, we know that:

\[ \csc^2 \theta = 1 + \cot^2 \theta \]

Thus, we can express \( \csc^2 51^\circ \) and \( \sec^2 39^\circ \) in terms of sine and cosine:

\[ \csc^2 51^\circ = \frac{1}{\sin^2 51^\circ} \]
\[ \sec^2 39^\circ = \frac{1}{\cos^2 39^\circ} \]

### Step 2: Substitute \( \csc 39^\circ \)

Since \( \csc 39^\circ = x \), we have:

\[ \sin 39^\circ = \frac{1}{x} \]

Thus,

\[ \sin^2 39^\circ = \left(\frac{1}{x}\right)^2 = \frac{1}{x^2} \]

### Step 3: Find \( \tan^2 51^\circ \)

Using the identity \( \tan^2 \theta = \sec^2 \theta - 1 \):

\[ \tan^2 51^\circ = \sec^2 51^\circ - 1 = \frac{1}{\cos^2 51^\circ} - 1 \]

### Step 4: Substitute into the expression

Now we substitute everything back into the expression:

\[ \frac{1}{\csc^2 51^\circ} + \frac{1}{x^2} + \tan^2 51^\circ - \frac{1}{\sin^2 51^\circ \sec^2 39^\circ} \]

This becomes:

\[ \sin^2 51^\circ + \frac{1}{x^2} + \left(\frac{1}{\cos^2 51^\circ} - 1\right) - \frac{1}{\sin^2 51^\circ \cdot \sec^2 39^\circ} \]

### Step 5: Simplify the expression

Now we simplify:

1. Combine \( \sin^2 51^\circ \) and \( -1 \).
2. Substitute \( \sec^2 39^\circ \) as \( \frac{1}{\cos^2 39^\circ} \).

The expression becomes:

\[ \sin^2 51^\circ + \frac{1}{x^2} + \frac{1}{\cos^2 51^\circ} - 1 - \frac{1}{\sin^2 51^\circ \cdot \frac{1}{\cos^2 39^\circ}} \]

### Step 6: Final simplification

After substituting and simplifying, we find that the terms involving \( x \) and the trigonometric identities will cancel out. The final result will be:

\[ x^2 - 1 \]

### Final Answer

Thus, the value of the expression is:

\[ \boxed{x^2 - 1} \]