The arithmetic mean of the numbers 2sin 2 ^ o , 4sin 4 ^ o , 6sin 6...

Total number of given numbers $= 90$
Sum of numbers,
$S = 2sin {2}^{o} + 4sin {4}^{o} + . \ldots \ldots . . + 178sin {178}^{o} + 180sin {180}^{o}$
$S = 178sin {2}^{o} + 176sin {4}^{o} + . \ldots \ldots . . + 2sin {178}^{o}$
Adding both the equation, we get,
$2S = 180\left(sin {2}^{o} + sin {4}^{o} + . \ldots \ldots . . + sin {178}^{o}\right) + 0$
$S = 90 \left(sin {2}^{o} + sin {4}^{o} + . \ldots \ldots . . + sin {178}^{o}\right)$
$⇒$ Mean $= \dfrac{S}{90} =$ sin {2}^{o} + sin {4}^{o} + . \ldots \ldots + sin {178}^{o}$<br />=\dfrac{sin {89}^{o}}{sin {1}^{o}}$ \sin \left(\dfrac{{2}^{o} + {178}^{o}}{2}\right)$<br />$= \dfrac{cos {1}^{o}}{sin {1}^{o}} \timessin {90}^{o} = cot {1}^{o}