The value of `cottheta.tan(90^(@)-theta)-sec(90^(@)-theta)cosectheta+(sin^(2)25^(@)+sin^(2)65^(@))` is

Options

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

Answer

To solve the expression \( \cot \theta \cdot \tan(90^\circ - \theta) - \sec(90^\circ - \theta) \cdot \csc \theta + (\sin^2 25^\circ + \sin^2 65^\circ) \), we will follow these steps:

### Step 1: Simplify \( \tan(90^\circ - \theta) \)
Using the identity \( \tan(90^\circ - \theta) = \cot \theta \), we can rewrite the expression:
\[ \cot \theta \cdot \tan(90^\circ - \theta) = \cot \theta \cdot \cot \theta = \cot^2 \theta \]

### Step 2: Simplify \( \sec(90^\circ - \theta) \)
Using the identity \( \sec(90^\circ - \theta) = \csc \theta \), we can rewrite the second part:
\[ -\sec(90^\circ - \theta) \cdot \csc \theta = -\csc \theta \cdot \csc \theta = -\csc^2 \theta \]

### Step 3: Combine the simplified parts
Now we can combine the parts we simplified:
\[ \cot^2 \theta - \csc^2 \theta + (\sin^2 25^\circ + \sin^2 65^\circ) \]

### Step 4: Simplify \( \sin^2 25^\circ + \sin^2 65^\circ \)
Using the identity \( \sin(90^\circ - \theta) = \cos \theta \), we have:
\[ \sin^2 65^\circ = \cos^2 25^\circ \]
Thus,
\[ \sin^2 25^\circ + \sin^2 65^\circ = \sin^2 25^\circ + \cos^2 25^\circ = 1 \]

### Step 5: Substitute back into the expression
Now we substitute back into our expression:
\[ \cot^2 \theta - \csc^2 \theta + 1 \]

### Step 6: Use the identity \( \csc^2 \theta = 1 + \cot^2 \theta \)
Using the identity \( \csc^2 \theta = 1 + \cot^2 \theta \), we can rewrite:
\[ \cot^2 \theta - (1 + \cot^2 \theta) + 1 = \cot^2 \theta - 1 - \cot^2 \theta + 1 \]

### Step 7: Simplify the expression
This simplifies to:
\[ 0 \]

### Final Answer
Thus, the value of the expression is:
\[ \boxed{0} \]