The average value of `sin2^@,sin4^@, sin6^@.........sin180^@` is `(i) 1/90 cos1^0 (ii) 1/90 sin1^0 (iii) 1/90cot1^0` (iv) none of these

Explanation

To find the average value of \( \sin 2^\circ, \sin 4^\circ, \sin 6^\circ, \ldots, \sin 180^\circ \), we will follow these steps:

### Step 1: Identify the terms in the series
The series consists of the sine of even angles from \( 2^\circ \) to \( 180^\circ \). The terms can be represented as:
\[ \sin 2^\circ, \sin 4^\circ, \sin 6^\circ, \ldots, \sin 180^\circ \]
This series has a total of \( 90 \) terms because the angles increase by \( 2^\circ \) from \( 2^\circ \) to \( 180^\circ \).

### Step 2: Calculate the sum of the terms
We can pair the terms in the series:
\[ \sin 2^\circ + \sin 178^\circ, \sin 4^\circ + \sin 176^\circ, \ldots, \sin 88^\circ + \sin 92^\circ \]
Using the identity \( \sin(180^\circ - x) = \sin x \), we find that:
\[ \sin 178^\circ = \sin 2^\circ, \sin 176^\circ = \sin 4^\circ, \ldots, \sin 92^\circ = \sin 88^\circ \]
Thus, each pair sums to \( 2\sin x \) for \( x = 2^\circ, 4^\circ, \ldots, 88^\circ \).

### Step 3: Count the pairs
There are \( 44 \) pairs (from \( \sin 2^\circ \) to \( \sin 88^\circ \)) plus the middle term \( \sin 90^\circ \):
\[ \sin 90^\circ = 1 \]
So the total sum becomes:
\[ \text{Sum} = 2(\sin 2^\circ + \sin 4^\circ + \sin 6^\circ + \ldots + \sin 88^\circ) + 1 \]

### Step 4: Use a sine summation formula
The sum \( \sin 2^\circ + \sin 4^\circ + \ldots + \sin 88^\circ \) can be computed using the formula:
\[ \sum_{k=1}^{n} \sin a + (k-1)d = \frac{\sin\left(\frac{nd}{2}\right) \sin\left(a + \frac{(n-1)d}{2}\right)}{\sin\left(\frac{d}{2}\right)} \]
For our case, \( a = 2^\circ \), \( d = 2^\circ \), and \( n = 44 \).

### Step 5: Calculate the average
The average value \( m \) is given by:
\[ m = \frac{\text{Sum}}{90} \]
Substituting the calculated sum into this formula gives us the average.

### Step 6: Final simplification
After simplification, we find:
\[ m = \frac{1}{90} \cot 1^\circ \]
Thus, the average value of \( \sin 2^\circ, \sin 4^\circ, \ldots, \sin 180^\circ \) is:
\[ \boxed{\frac{1}{90} \cot 1^\circ} \]