Question

If `cosx-sinalphacotbetasinx=cosa ,` then the value of `tan(x/2)` is (a)`-tan(alpha/2)cot(beta/2)` (b) `tan(alpha/2)tan(beta/2)` (c)`-cot((alphabeta)/2)tan(beta/2)` (d) `cot(alpha/2)cot(beta/2)`

Answer 1

`cosx-sinalphacotbetasinx = cosalpha`
`=>(1-tan^2(x/2))/(1+tan^2(x/2)) - sinalphacotbeta((2tan(x/2))/(1-tan^2(x/2))) = cosalpha`
`=>1-tan^2(x/2) - sinalphacotbeta(2tan(x/2)) = cosalpha - cosalphatan^2(x/2)`
`=>tan^2(x/2)(1+cosalpha) +2sinalphacotbetatan(x/2) - (1-cosalpha) = 0`
`=>tan^2(x/2)(1+cosalpha) +2sinalpha(cosbeta/sinbeta)tan(x/2) - (1-cosalpha) = 0`
`=>tan^2(x/2)(1+cosalpha) +2sinalpha((cos^2(beta/2)-sin^2(beta/2))/(2sin(beta/2)cos(beta/2)))tan(x/2) - (1-cosalpha) = 0`
`=>tan^2(x/2)(1+cosalpha) +sinalpha(cot(beta/2)-tan(beta/2))tan(x/2) - (1-cosalpha) = 0`
`=>(tan(x/2)+cot(beta/2)tan(alpha/2))(tan(x/2)-tan(beta/2)tan(alpha/2)) = 0`
`=>(tan(x/2)+cot(beta/2)tan(alpha/2)) = 0 or (tan(x/2)-tan(beta/2)tan(alpha/2)) = 0`
`=>tan(x/2) = - cot(beta/2)tan(alpha/2) or tan(x/2) = tan(beta/2)tan(alpha/2)`
So, options `(a)` and `(b)` are the correct options.