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The value of (cot84^(@)cot48^(@))/(cot66...

The value of `(cot84^(@)cot48^(@))/(cot66^(@)cot78^(@))` is equal to

A

1

B

0

C

`sqrt(3)`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
A
`(cot 84^(@)cot 48^(@))/(cot 66^(@)cot 78^(@))`
`=((2 cos 84^(@)sin 66^(@))(2sin 78^(@)cos 48^(@)))/((2 sin 84^(@)cos 66^(@))(2 cos 78^(@)sin 48^(@)))`
`=((sin 150^(@)-sin 18^(@))(sin 126^(@)+sin 30^(@)))/((sin 150^(@)+sin 18^(@))(sin 126^(@)-sin 30^(@)))`
`=((sin 30^(@)-sin 18^(@))(sin 54^(@)+sin 30^(@)))/((sin 30^(@)+sin 18^(@))(sin 54^(@)-sin 30^(@)))`
`=(((1)/(2)-(sqrt(5)-1)/(4))((sqrt(5)+1)/(4)+(1)/(2)))/(((1)/(2)+(sqrt(5)-1)/(4))((sqrt(5)+1)/(4)-(1)/(2)))`
`=((3-sqrt(5))(3+sqrt(5)))/((sqrt(5)+1)(sqrt(5)-1))=(9-5)/(5-1)=(4)/(4)=1`
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