Which of the following expressions is not equal to sin `(-135^@)` ?

Options

A. sin `(135^@)`
B. cos `(135^@)`
C. `-cos (-45^@)`
D. `sin (225^@)`

Answer

To determine which of the following expressions is not equal to \(\sin(-135^\circ)\), we will first calculate \(\sin(-135^\circ)\) and then evaluate the given options one by one.

### Step 1: Calculate \(\sin(-135^\circ)\)

Using the identity for sine of a negative angle:
\[ \sin(-x) = -\sin(x) \]
we can write:
\[ \sin(-135^\circ) = -\sin(135^\circ) \]

### Step 2: Calculate \(\sin(135^\circ)\)

Next, we can express \(135^\circ\) as:
\[ 135^\circ = 90^\circ + 45^\circ \]
Using the sine addition formula:
\[ \sin(90^\circ + x) = \cos(x) \]
we find:
\[ \sin(135^\circ) = \sin(90^\circ + 45^\circ) = \cos(45^\circ) = \frac{1}{\sqrt{2}} \]

### Step 3: Substitute back to find \(\sin(-135^\circ)\)

Now substituting back, we have:
\[ \sin(-135^\circ) = -\sin(135^\circ) = -\frac{1}{\sqrt{2}} \]

### Step 4: Evaluate the options

Now we will evaluate each option to see which one is not equal to \(-\frac{1}{\sqrt{2}}\).

1. **Option 1: \(\sin(135^\circ)\)**
\[ \sin(135^\circ) = \frac{1}{\sqrt{2}} \quad \text{(This is equal to } -\frac{1}{\sqrt{2}} \text{)} \]

2. **Option 2: \(\cos(135^\circ)\)**
\[ \cos(135^\circ) = \cos(90^\circ + 45^\circ) = -\sin(45^\circ) = -\frac{1}{\sqrt{2}} \quad \text{(This is equal to } -\frac{1}{\sqrt{2}} \text{)} \]

3. **Option 3: \(-\cos(-45^\circ)\)**
\[ -\cos(-45^\circ) = -\cos(45^\circ) = -\frac{1}{\sqrt{2}} \quad \text{(This is equal to } -\frac{1}{\sqrt{2}} \text{)} \]

4. **Option 4: \(\sin(225^\circ)\)**
\[ \sin(225^\circ) = \sin(360^\circ - 135^\circ) = -\sin(135^\circ) = -\frac{1}{\sqrt{2}} \quad \text{(This is equal to } -\frac{1}{\sqrt{2}} \text{)} \]

### Conclusion

From the evaluations, we see that:
- Option 1: \(\sin(135^\circ) = \frac{1}{\sqrt{2}}\) is **not equal** to \(-\frac{1}{\sqrt{2}}\).
- Options 2, 3, and 4 are all equal to \(-\frac{1}{\sqrt{2}}\).

Thus, the expression that is not equal to \(\sin(-135^\circ)\) is:
\[ \text{Option 1: } \sin(135^\circ) \]