What is the value of `[tan^(2)(90-theta)-sin^(2)(90-theta)]cosec^(2)(90-theta)cot^(2)(90-theta)` ?

Options

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

Answer

To solve the expression \([tan^2(90^\circ - \theta) - sin^2(90^\circ - \theta)] \cdot cosec^2(90^\circ - \theta) \cdot cot^2(90^\circ - \theta)\), we will follow these steps:

### Step 1: Use Trigonometric Identities
We know the following trigonometric identities:
- \(tan(90^\circ - \theta) = cot(\theta)\)
- \(sin(90^\circ - \theta) = cos(\theta)\)
- \(cosec(90^\circ - \theta) = sec(\theta)\)
- \(cot(90^\circ - \theta) = tan(\theta)\)

Using these identities, we can rewrite the expression:

\[ tan^2(90^\circ - \theta) = cot^2(\theta) \]
\[ sin^2(90^\circ - \theta) = cos^2(\theta) \]
\[ cosec^2(90^\circ - \theta) = sec^2(\theta) \]
\[ cot^2(90^\circ - \theta) = tan^2(\theta) \]

### Step 2: Substitute the Identities into the Expression
Now substituting these into the original expression, we have:

\[ [cot^2(\theta) - cos^2(\theta)] \cdot sec^2(\theta) \cdot tan^2(\theta) \]

### Step 3: Simplify the Expression
Next, we can simplify the expression inside the brackets:

\[ cot^2(\theta) = \frac{cos^2(\theta)}{sin^2(\theta)} \]

Thus, the expression becomes:

\[ \left[\frac{cos^2(\theta)}{sin^2(\theta)} - cos^2(\theta)\right] \cdot sec^2(\theta) \cdot tan^2(\theta) \]

Finding a common denominator inside the brackets:

\[ \left[\frac{cos^2(\theta) - cos^2(\theta) \cdot sin^2(\theta)}{sin^2(\theta)}\right] \cdot sec^2(\theta) \cdot tan^2(\theta) \]

### Step 4: Factor Out the Common Terms
Factoring out \(cos^2(\theta)\):

\[ \frac{cos^2(\theta)(1 - sin^2(\theta))}{sin^2(\theta)} \cdot sec^2(\theta) \cdot tan^2(\theta) \]

Using the identity \(1 - sin^2(\theta) = cos^2(\theta)\):

\[ \frac{cos^2(\theta) \cdot cos^2(\theta)}{sin^2(\theta)} \cdot sec^2(\theta) \cdot tan^2(\theta) \]

### Step 5: Substitute Back the Trigonometric Functions
Now substituting back \(sec^2(\theta) = \frac{1}{cos^2(\theta)}\) and \(tan^2(\theta) = \frac{sin^2(\theta)}{cos^2(\theta)}\):

The expression simplifies to:

\[ \frac{cos^4(\theta)}{sin^2(\theta)} \cdot \frac{1}{cos^2(\theta)} \cdot \frac{sin^2(\theta)}{cos^2(\theta)} \]

### Step 6: Cancel Out Terms
Now we can cancel out terms:

\[ \frac{cos^4(\theta) \cdot sin^2(\theta)}{sin^2(\theta) \cdot cos^2(\theta) \cdot cos^2(\theta)} = 1 \]

### Final Answer
Thus, the value of the expression is:

\[ \boxed{1} \]