If `alpha and beta` are non-zero real number such that `2(cos beta-cos alpha)+cos alpha cos beta=1.` Then which of the following is treu?

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)`