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If theta = tan^(-1)(2/9) + tan^(-1) (4/3...

If `theta = tan^(-1)(2/9) + tan^(-1) (4/33) + tan^(-1)(8/128) + tan^(-1)(16/513) +……….` upton `infty`, then `cot theta`=

A

1

B

2

C

`sqrt(3)`

D

`1/sqrt(3)`

Text Solution

Verified by Experts

The correct Answer is:
B
`theta = tan^(-1) 2/(1+4 xx 2) + tan^(-1) 4/(1+8 xx 4) + tan^(-1) 8/(1+16 xx 8) + tan^(-1) 16/(1+ 32 xx 16) +….. = tan^(-1) ((4-2)/(1+4 xx 2)) + tan^(-1)(8-4)/(1+ 8 xx 4) + tan^(-1)((16-8)/(1+ 16 xx 8)) + tan^(-1) ((32 - 16)/(1+ 32 xx 16)) +…..`
`=(tan^(-1)4 - tan^(-1)2) + (tan^(-1)8 - tan^(-1)4) + (tan^(-1)16 - tan^(-1)8) + (tan^(-1)16 - tan^(-1)8) + (tan^(-1) 32 - tan^(-1) 16) +.... to infty`
`pi/2 - tan^(-1)2`
`rArr cot(pi/2 - tan^(-1)2) = cot(cot^(-1)2)=2`
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