The value of `cos (270^(@) + theta) cos ( 90^(@) -theta) -sin (270^(@) - theta) cos theta ` is

Options

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

Explanation

To solve the expression \( \cos(270^\circ + \theta) \cos(90^\circ - \theta) - \sin(270^\circ - \theta) \cos \theta \), we will break it down step by step.

### Step 1: Evaluate \( \cos(270^\circ + \theta) \)
Using the cosine addition formula and the properties of trigonometric functions:
\[ \cos(270^\circ + \theta) = -\sin(\theta) \]
This is because \( 270^\circ \) is in the third quadrant where cosine is negative, and it shifts the sine function.

### Step 2: Evaluate \( \cos(90^\circ - \theta) \)
Using the co-function identity:
\[ \cos(90^\circ - \theta) = \sin(\theta) \]

### Step 3: Substitute values into the expression
Now substituting the values from Steps 1 and 2 into the original expression:
\[ \cos(270^\circ + \theta) \cos(90^\circ - \theta) = (-\sin(\theta))(\sin(\theta)) = -\sin^2(\theta) \]

### Step 4: Evaluate \( \sin(270^\circ - \theta) \)
Using the sine subtraction formula:
\[ \sin(270^\circ - \theta) = -\cos(\theta) \]
This is because \( 270^\circ \) is in the third quadrant where sine is negative.

### Step 5: Substitute \( \sin(270^\circ - \theta) \) into the expression
Now substituting this value into the expression:
\[ -\sin(270^\circ - \theta) \cos \theta = -(-\cos(\theta)) \cos \theta = \cos^2(\theta) \]

### Step 6: Combine the results
Now we combine the results from Steps 3 and 5:
\[ -\sin^2(\theta) + \cos^2(\theta) \]

### Step 7: Use the Pythagorean identity
Using the Pythagorean identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \):
\[ -\sin^2(\theta) + \cos^2(\theta) = \cos^2(\theta) - \sin^2(\theta) \]
This can be rewritten as:
\[ \cos^2(\theta) - \sin^2(\theta) = 1 \]

### Final Result
Thus, the value of the expression is:
\[ \cos^2(\theta) - \sin^2(\theta) = 1 \]

### Conclusion
The final answer is:
\[ \boxed{1} \]