Let A and B denote the statements A: `"cos"a+"cos"b+"cos"g=""0` B : `"sin"a+"sin"b+"sin"g=""0` If `cos(b g)""+""cos(g a)""+""cos(a b)""=""""3//2` , then (1) A is true and B is false (2) A is false and B is true (3) both A and B are true (4) both A and B are false

Explanation

`cos(beta- gamma) + cos(gamma- alpha) + cos(alpha- beta) = -3/2`

`2cos(beta- gamma) + 2 cos ( gamma- alpha) + 2 cos( alpha - beta) + 3= 0`

`2 cos( beta- gamma) + 2cos( gamma - alpha) + 2cos(alpha- beta) + sin^2 alpha + cos^2 alpha + sin^2 beta + cos^2 beta + sin^2 gamma + cos^2 gamma = 0`

now, using identity, `(a+b+c)^2 = a^2 + b^2 + c^2 + 2(ab + bc+ ac)`

`(cos alpha + cos beta + cos gamma)^2 + (sin alpha + sin beta + sin gamma)^2 = 0`

`= 0+ 0 = 0`

so, option 3 is correct