If `x=sec57^(@)`, then
`cot^(2)33^(@)+sin^(2)57^(@)+sin^(2)33^(@)+cosec^(2)57^(@)cos^(2)33^(@)+sec^(2)33^(@)sin^(2)57^(@)` is equal to:

Options

A. `2x^(2)+1`
B. `x^(2)+1`
C. `(1)/(x^(2)+1)`
D. `x^(2)+2`

Answer

To solve the expression given in the question, we will follow a systematic approach.

Given:
\[ x = \sec 57^\circ \]

We need to evaluate:
\[ \cot^2 33^\circ + \sin^2 57^\circ + \sin^2 33^\circ + \csc^2 57^\circ \cos^2 33^\circ + \sec^2 33^\circ \sin^2 57^\circ \]

### Step 1: Use Trigonometric Identities
We know the following trigonometric identities:
1. \( \sin^2 \theta + \cos^2 \theta = 1 \)
2. \( \csc^2 \theta = 1 + \cot^2 \theta \)
3. \( \sec^2 \theta = 1 + \tan^2 \theta \)

### Step 2: Substitute Values
We can start substituting the known values:
- \( \sin^2 57^\circ + \cos^2 57^\circ = 1 \)
- \( \sin^2 33^\circ + \cos^2 33^\circ = 1 \)

### Step 3: Rewrite the Expression
Now, rewrite the expression using the identities:
1. Replace \( \csc^2 57^\circ \) with \( 1 + \cot^2 57^\circ \)
2. Replace \( \sec^2 33^\circ \) with \( 1 + \tan^2 33^\circ \)

Thus, the expression becomes:
\[ \cot^2 33^\circ + \sin^2 57^\circ + \sin^2 33^\circ + (1 + \cot^2 57^\circ) \cos^2 33^\circ + (1 + \tan^2 33^\circ) \sin^2 57^\circ \]

### Step 4: Simplify the Expression
Now, we can simplify the expression step by step:
- Combine like terms.
- Use \( \sin^2 57^\circ + \cos^2 57^\circ = 1 \) and \( \sin^2 33^\circ + \cos^2 33^\circ = 1 \).

### Step 5: Final Calculation
After simplification, we will arrive at:
\[ \text{Final Expression} = 3 + \cot^2 33^\circ \]

### Step 6: Substitute for \( x \)
Since \( x = \sec 57^\circ \), we know:
\[ \sec^2 57^\circ = 1 + \tan^2 57^\circ \]
Using the relationship between secant and tangent, we can relate \( \cot^2 33^\circ \) to \( x \).

### Conclusion
Thus, the final result can be expressed in terms of \( x \):
\[ \text{Final Answer} = x^2 + 2 \]