The value of ` (sin 55^(@) - cos 55^(@))/(sin10^(@)` is

Options

A. `(1)/(sqrt2)`
B. 2
C. 1
D. `sqrt(2)`

Answer

To solve the expression \( \frac{\sin 55^\circ - \cos 55^\circ}{\sin 10^\circ} \), we can follow these steps:

### Step 1: Rewrite \( \sin 55^\circ \)
We can use the co-function identity for sine:
\[ \sin 55^\circ = \cos(90^\circ - 55^\circ) = \cos 35^\circ \]
Thus, we rewrite the expression:
\[ \frac{\sin 55^\circ - \cos 55^\circ}{\sin 10^\circ} = \frac{\cos 35^\circ - \cos 55^\circ}{\sin 10^\circ} \]

### Step 2: Use the cosine difference identity
We can apply the cosine difference formula:
\[ \cos x - \cos y = -2 \sin\left(\frac{x+y}{2}\right) \sin\left(\frac{x-y}{2}\right) \]
Here, let \( x = 35^\circ \) and \( y = 55^\circ \):
\[ \cos 35^\circ - \cos 55^\circ = -2 \sin\left(\frac{35^\circ + 55^\circ}{2}\right) \sin\left(\frac{35^\circ - 55^\circ}{2}\right) \]

### Step 3: Calculate the averages
Calculating the averages:
\[ \frac{35^\circ + 55^\circ}{2} = \frac{90^\circ}{2} = 45^\circ \]
\[ \frac{35^\circ - 55^\circ}{2} = \frac{-20^\circ}{2} = -10^\circ \]
Thus, we have:
\[ \cos 35^\circ - \cos 55^\circ = -2 \sin(45^\circ) \sin(-10^\circ) \]

### Step 4: Simplify the expression
Using the fact that \( \sin(-\theta) = -\sin(\theta) \):
\[ \cos 35^\circ - \cos 55^\circ = -2 \sin(45^\circ)(-\sin(10^\circ)) = 2 \sin(45^\circ) \sin(10^\circ) \]
Now substituting this back into our expression:
\[ \frac{\cos 35^\circ - \cos 55^\circ}{\sin 10^\circ} = \frac{2 \sin(45^\circ) \sin(10^\circ)}{\sin(10^\circ)} \]

### Step 5: Cancel out \( \sin 10^\circ \)
Since \( \sin 10^\circ \) is in both the numerator and denominator, we can cancel it out:
\[ = 2 \sin(45^\circ) \]

### Step 6: Substitute the value of \( \sin(45^\circ) \)
We know that:
\[ \sin(45^\circ) = \frac{1}{\sqrt{2}} \]
Thus:
\[ 2 \sin(45^\circ) = 2 \cdot \frac{1}{\sqrt{2}} = \frac{2}{\sqrt{2}} = \sqrt{2} \]

### Final Answer
Therefore, the value of \( \frac{\sin 55^\circ - \cos 55^\circ}{\sin 10^\circ} \) is:
\[ \sqrt{2} \]