The value of ` sin^(2) (90^(@) - theta) [ 1 + cot ^(2) (90^(@) - theta)]` is

Options

A. `-1`
B. 0
C. `(1)/(2)`
D. 1

Answer

To solve the problem, we need to evaluate the expression:

\[ \sin^2(90^\circ - \theta) \left[ 1 + \cot^2(90^\circ - \theta) \right] \]

### Step 1: Use the co-function identity for sine
We know from trigonometric identities that:

\[ \sin(90^\circ - \theta) = \cos(\theta) \]

Thus,

\[ \sin^2(90^\circ - \theta) = \cos^2(\theta) \]

### Step 2: Use the co-function identity for cotangent
Similarly, we can use the identity for cotangent:

\[ \cot(90^\circ - \theta) = \tan(\theta) \]

Therefore,

\[ \cot^2(90^\circ - \theta) = \tan^2(\theta) \]

### Step 3: Substitute the identities into the expression
Now, substituting these identities into the original expression, we get:

\[ \sin^2(90^\circ - \theta) \left[ 1 + \cot^2(90^\circ - \theta) \right] = \cos^2(\theta) \left[ 1 + \tan^2(\theta) \right] \]

### Step 4: Use the identity for \(1 + \tan^2(\theta)\)
We know from trigonometric identities that:

\[ 1 + \tan^2(\theta) = \sec^2(\theta) \]

### Step 5: Substitute back into the expression
Thus, we can rewrite our expression as:

\[ \cos^2(\theta) \cdot \sec^2(\theta) \]

### Step 6: Simplify the expression
Since \(\sec(\theta) = \frac{1}{\cos(\theta)}\), we have:

\[ \sec^2(\theta) = \frac{1}{\cos^2(\theta)} \]

So,

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

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

\[ \boxed{1} \]

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