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 \([tan^2 (90 - \theta) - sin^2 (90 - \theta)] \cdot cosec^2 (90 - \theta) \cdot cot^2 (90 - \theta)\), we can follow these steps:

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

### Step 2: Substitute the Identities
Using the identities, we can rewrite the expression as follows:
\[ tan^2(90 - \theta) = cot^2(\theta) \]
\[ sin^2(90 - \theta) = cos^2(\theta) \]
\[ cosec^2(90 - \theta) = sec^2(\theta) \]
\[ cot^2(90 - \theta) = tan^2(\theta) \]

Thus, the expression becomes:
\[ [cot^2(\theta) - cos^2(\theta)] \cdot sec^2(\theta) \cdot tan^2(\theta) \]

### Step 3: Simplify the Expression
Now we can simplify the expression:
\[ cot^2(\theta) = \frac{cos^2(\theta)}{sin^2(\theta)} \]
So,
\[ cot^2(\theta) - cos^2(\theta) = \frac{cos^2(\theta)}{sin^2(\theta)} - cos^2(\theta) \]
To combine these fractions, we can express \(cos^2(\theta)\) with a common denominator:
\[ = \frac{cos^2(\theta) - cos^2(\theta) \cdot sin^2(\theta)}{sin^2(\theta)} = \frac{cos^2(\theta)(1 - sin^2(\theta))}{sin^2(\theta)} \]
Using the identity \(1 - sin^2(\theta) = cos^2(\theta)\), we get:
\[ = \frac{cos^4(\theta)}{sin^2(\theta)} \]

### Step 4: Substitute Back into the Expression
Now substitute this back into the expression:
\[ \frac{cos^4(\theta)}{sin^2(\theta)} \cdot sec^2(\theta) \cdot tan^2(\theta) \]
We know that:
\[ sec^2(\theta) = \frac{1}{cos^2(\theta)} \quad \text{and} \quad tan^2(\theta) = \frac{sin^2(\theta)}{cos^2(\theta)} \]

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

### Step 5: Simplify Further
Now simplifying:
\[ = \frac{cos^4(\theta) \cdot sin^2(\theta)}{sin^2(\theta) \cdot cos^4(\theta)} = 1 \]

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