What is the simplified value of `sin^(2) (90- theta) - [({sin(90-theta) sin theta})/(tan theta)]`?

Options

A. 1
B. cosec `theta`
C. 0
D. `cos theta`

Answer

To simplify the expression \( \sin^2(90^\circ - \theta) - \left( \frac{\sin(90^\circ - \theta) \sin \theta}{\tan \theta} \right) \), we can follow these steps:

### Step 1: Use the co-function identity for sine
We know that:
\[ \sin(90^\circ - \theta) = \cos(\theta) \]
Thus, we can rewrite the expression:
\[ \sin^2(90^\circ - \theta) = \cos^2(\theta) \]

### Step 2: Substitute into the expression
Now substitute \( \sin(90^\circ - \theta) \) in the expression:
\[ \cos^2(\theta) - \left( \frac{\cos(\theta) \sin(\theta)}{\tan(\theta)} \right) \]

### Step 3: Simplify the fraction
Recall that:
\[ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \]
Thus, we can rewrite the fraction:
\[ \frac{\cos(\theta) \sin(\theta)}{\tan(\theta)} = \frac{\cos(\theta) \sin(\theta)}{\frac{\sin(\theta)}{\cos(\theta)}} = \cos^2(\theta) \]

### Step 4: Substitute back into the expression
Now the expression becomes:
\[ \cos^2(\theta) - \cos^2(\theta) \]

### Step 5: Simplify the expression
This simplifies to:
\[ 0 \]

### Final Answer
Thus, the simplified value of the expression is:
\[ \boxed{0} \]