If `4nalpha =pi` then `cot alpha cot 2 alpha cot 3alpha ...cot (2n-1)alpha` `n in Z` is equal to

Answer

1

Explanation

We have,
`4nalpha=pi`
`implies2nalpha=(pi)/(2)`
`implies2nalpha-alpha=(pi)/(2)-alpha,2n alpha-2alpha=(pi)/(2)-2alpha, 2nalpha-3alpha=(pi)/(2)-3alpha` and so on.
`impliescot(2n-1)alpha=tan alpha, cot(2n-2)alpha=tan2 alphaand` so on.
`thereforecot alphacot 2 alphacot 3 alpha... cot(2n-1)alpha`
`=cot alpha cot2 alpha cot 3 alpha... tan2 alpha. tan alpha`
`=(cotalpha.tan alpha)(cot2alpha. tan 2 alpha)(cot 3 alpha. tan3 alpha)...=1`