What is the value of `[1-tan(90^(@)-theta)]^(2)*[cos^(2)(90^(@)-theta) -1]`?

Options

A. `Cot^2theta(sin2theta`-1)
B. `-cos2theta`
C. `cos2theta`
D. `sin2theta`

Answer

To solve the expression \([1 - \tan(90^\circ - \theta)]^2 \cdot [\cos^2(90^\circ - \theta) - 1]\), we will use trigonometric identities. Let's break it down step by step.

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

**Hint:** Remember that \(\tan(90^\circ - \theta) = \cot(\theta)\).

### Step 2: Simplify \(\cos(90^\circ - \theta)\)
Using the co-function identity for cosine:
\[ \cos(90^\circ - \theta) = \sin(\theta) \]
Now, we can rewrite the second part of the expression:
\[ \cos^2(90^\circ - \theta) - 1 = \sin^2(\theta) - 1 \]

**Hint:** Recall that \(\cos(90^\circ - \theta) = \sin(\theta)\).

### Step 3: Substitute back into the expression
Now substituting back into the expression, we have:
\[ [1 - \cot(\theta)]^2 \cdot [\sin^2(\theta) - 1] \]

### Step 4: Simplify \(\sin^2(\theta) - 1\)
Using the Pythagorean identity:
\[ \sin^2(\theta) - 1 = -\cos^2(\theta) \]
So, we can rewrite the expression as:
\[ [1 - \cot(\theta)]^2 \cdot [-\cos^2(\theta)] \]

**Hint:** Remember that \(\sin^2(\theta) + \cos^2(\theta) = 1\).

### Step 5: Expand \([1 - \cot(\theta)]^2\)
Now, let's expand \([1 - \cot(\theta)]^2\):
\[ [1 - \cot(\theta)]^2 = 1 - 2\cot(\theta) + \cot^2(\theta) \]

### Step 6: Combine the results
Now, substituting this back into our expression:
\[ (1 - 2\cot(\theta) + \cot^2(\theta)) \cdot [-\cos^2(\theta)] \]
Distributing \(-\cos^2(\theta)\):
\[ -\cos^2(\theta) + 2\cot(\theta)\cos^2(\theta) - \cot^2(\theta)\cos^2(\theta) \]

### Step 7: Final expression
Thus, the final expression simplifies to:
\[ -\cos^2(\theta) + 2\cot(\theta)\cos^2(\theta) - \cot^2(\theta)\cos^2(\theta) \]

### Conclusion
The value of the original expression \([1 - \tan(90^\circ - \theta)]^2 \cdot [\cos^2(90^\circ - \theta) - 1]\) simplifies to:
\[ -\cos^2(\theta) + 2\cot(\theta)\cos^2(\theta) - \cot^2(\theta)\cos^2(\theta) \]