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Let alpha and beta be non-zero real num...

Let `alpha and beta ` be non-zero real numbers such that ` 2 ( cos beta - cos alpha) + cos alpha cos beta =1`. Then which of the following is/are true ?

A

`sqrt(3) tan ((alpha)/(2))-tan((beta)/(2))=0`

B

`tan((alpha)/(2)) - sqrt(3) tan((beta)/(2)) = 0`

C

`tan((alpha)/(2)) + sqrt(3)((beta)/(2))=0`

D

`sqrt(3) tan((alpha)/(2))+ tan((beta)/(2))=0`

Text Solution

Verified by Experts

The correct Answer is:
B, C
We have, `2(cos beta - cos alpha)+ cos alpha cos beta = 1`
or `4(cos beta - cos alpha) + 2 cos alpha cos beta = 2`
`rArr 1-cos alpha + cos beta - cos alpha cos beta = 2`
`= 3+3 cos alpha - 3 cos beta - 3cos alpha cos beta`
`rArr (1-cos alpha) (1+ cos beta)=3(1+cos alpha) (1-cos beta)`
`rArr ((1-cos alpha))/((1+cos alpha)) = (3(1-cos beta))(1+cos beta))`
`rArr tan^(2) (alpha)/(2) = 3 tan^(2)(beta)/(2)`
`therefore tan.(alpha)/(2) pm sqrt(3) tan.(beta)/(2) =0`
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