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If 4 n alpha= pi, then cot alpha cot 2al...

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

A

1

B

`-1`

C

`oo`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
A
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`
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