What is the value of ` [ sin ( 90^(@) - 10 theta) - cos (pi - 6 theta) ] / [ cos ( pi/2 - 10 theta)- sin (pi - 6 theta) ]` ?

Options

A. `tan 2 theta`
B. `cot 2 theta`
C. ` cot theta`
D. `cot 3 theta`

Answer

To solve the expression

\[ \frac{ \sin(90^\circ - 10\theta) - \cos(\pi - 6\theta) }{ \cos\left(\frac{\pi}{2} - 10\theta\right) - \sin(\pi - 6\theta) } \]

we can simplify it step by step.

### Step 1: Simplify the numerator
Using the identity \(\sin(90^\circ - x) = \cos(x)\), we have:

\[ \sin(90^\circ - 10\theta) = \cos(10\theta) \]

Now, for \(\cos(\pi - x) = -\cos(x)\):

\[ \cos(\pi - 6\theta) = -\cos(6\theta) \]

Thus, the numerator becomes:

\[ \cos(10\theta) - (-\cos(6\theta)) = \cos(10\theta) + \cos(6\theta) \]

### Step 2: Simplify the denominator
Using the identity \(\cos\left(\frac{\pi}{2} - x\right) = \sin(x)\):

\[ \cos\left(\frac{\pi}{2} - 10\theta\right) = \sin(10\theta) \]

And again using \(\sin(\pi - x) = \sin(x)\):

\[ \sin(\pi - 6\theta) = \sin(6\theta) \]

Thus, the denominator becomes:

\[ \sin(10\theta) - \sin(6\theta) \]

### Step 3: Rewrite the expression
Now we can rewrite the entire expression:

\[ \frac{ \cos(10\theta) + \cos(6\theta) }{ \sin(10\theta) - \sin(6\theta) } \]

### Step 4: Use trigonometric identities
We can use the sum-to-product identities:

1. For the numerator:
\[ \cos(10\theta) + \cos(6\theta) = 2 \cos\left(\frac{10\theta + 6\theta}{2}\right) \cos\left(\frac{10\theta - 6\theta}{2}\right) = 2 \cos(8\theta) \cos(2\theta) \]

2. For the denominator:
\[ \sin(10\theta) - \sin(6\theta) = 2 \cos\left(\frac{10\theta + 6\theta}{2}\right) \sin\left(\frac{10\theta - 6\theta}{2}\right) = 2 \cos(8\theta) \sin(2\theta) \]

### Step 5: Substitute back into the expression
Now substituting back into the expression gives us:

\[ \frac{ 2 \cos(8\theta) \cos(2\theta) }{ 2 \cos(8\theta) \sin(2\theta) } \]

### Step 6: Simplify the expression
The \(2 \cos(8\theta)\) cancels out (assuming \(\cos(8\theta) \neq 0\)), leading to:

\[ \frac{\cos(2\theta)}{\sin(2\theta)} = \cot(2\theta) \]

### Final Answer
Thus, the value of the original expression is:

\[ \cot(2\theta) \]