The numerical value of `1+(1)/(cot^(2)63^(@))-sec^(2)27^(@)+(1)/(sin^(2)63^(@))-"cosec"^(2)27^(@)` is

Options

A. 1
B. 2
C. `-1`
D. 0

Answer

To solve the expression \( 1 + \frac{1}{\cot^2 63^\circ} - \sec^2 27^\circ + \frac{1}{\sin^2 63^\circ} - \csc^2 27^\circ \), we will break it down step by step.

### Step 1: Rewrite the cotangent and secant functions
We know that:
\[ \cot^2 \theta = \frac{1}{\tan^2 \theta} \quad \text{and} \quad \sec^2 \theta = 1 + \tan^2 \theta \]
Thus,
\[ \frac{1}{\cot^2 63^\circ} = \tan^2 63^\circ \]
and
\[ \sec^2 27^\circ = 1 + \tan^2 27^\circ \]

### Step 2: Substitute into the expression
Substituting these identities into the expression, we have:
\[ 1 + \tan^2 63^\circ - (1 + \tan^2 27^\circ) + \frac{1}{\sin^2 63^\circ} - \csc^2 27^\circ \]
This simplifies to:
\[ \tan^2 63^\circ - \tan^2 27^\circ + \frac{1}{\sin^2 63^\circ} - \csc^2 27^\circ \]

### Step 3: Rewrite the sine and cosecant functions
We know that:
\[ \csc^2 \theta = \frac{1}{\sin^2 \theta} \]
Thus,
\[ \csc^2 27^\circ = \frac{1}{\sin^2 27^\circ} \]
Now, substituting this into the expression gives:
\[ \tan^2 63^\circ - \tan^2 27^\circ + \frac{1}{\sin^2 63^\circ} - \frac{1}{\sin^2 27^\circ} \]

### Step 4: Use the identity for tangent
Using the identity \(\tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta}\), we can rewrite \(\tan^2 63^\circ\) and \(\tan^2 27^\circ\):
\[ \tan^2 63^\circ = \frac{\sin^2 63^\circ}{\cos^2 63^\circ} \quad \text{and} \quad \tan^2 27^\circ = \frac{\sin^2 27^\circ}{\cos^2 27^\circ} \]
Substituting these into the expression gives:
\[ \frac{\sin^2 63^\circ}{\cos^2 63^\circ} - \frac{\sin^2 27^\circ}{\cos^2 27^\circ} + \frac{1}{\sin^2 63^\circ} - \frac{1}{\sin^2 27^\circ} \]

### Step 5: Combine the fractions
Now we can combine the fractions:
\[ \frac{\sin^2 63^\circ \sin^2 27^\circ - \sin^2 27^\circ \sin^2 63^\circ}{\cos^2 63^\circ \cos^2 27^\circ} + \left(\frac{\cos^2 27^\circ - \cos^2 63^\circ}{\sin^2 63^\circ \sin^2 27^\circ}\right) \]
This simplifies to:
\[ 0 + 0 = 0 \]

### Final Result
Thus, the numerical value of the expression is:
\[ \boxed{1} \]