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 ?

Options

A. `(alpha)/(2)=+-sqrt3tan""(beta)/(2)`
B. `tan""(beta)/(2)=pmsqrt3tan""(alpha)/(2)`
C. `tan""(alpha)/(2)=pmtan""(beta)/(2)`
D. `tan""(alpha)/(2)=pmsqrt2tan""(beta)/(2)`

Answer

A

Explanation

`impliescos beta(2+cosalpha)=1+2 cosalpha`
`impliescos beta=(1+2 cos alpha)/(2+cosalpha)`
`implies(cosbeta+1)/(cosbeta-1)=(2(1+cosalpha))/(cos alpha-1)`
`implies(1-cos beta)/(1+cos beta)=1/3((1-cos alpha)/(1+cosalpha))`
`impliestan^(2)""(beta)/(2)=1/3tan^(2)""(alpha)/(2)impliestan""(alpha)/(2)=pmsqrt3tan""(beta)/(2)`