Correct Answer – A::B
`cos x – sin alpha cos beta sinx = cos alpha `
`rArr (1-tan^(2)(x//2))/(1+tan^(2)(x//2)) – sin alpha cos beta (2tan(x//2))/(1+tan^(2)(x//2)) = cos alpha`
`rArr tan^(2) “”(x)/(2) (1+ cosalpha ) + 2 sin alpha cos beta tan “”(x)/(2) – (1- cos alpha) =0`
`rArr tan ^(2)””(x)/(2) + (2sinalpha cos beta)/(1+ cos alpha) tan””(x)/(2) – (1-cos alpha)/( 1+cos alpha) =0`
`rArr tan^(2)””(x)/(2) + 2 tan””(alpha)/(2) cos beta tan “”(x)/(2) – tan^(2) “”(alpha)/(2)=0`
`rArr tan^(2″”(x)/(2) + 2 tan””(alpha)/(2)*(1)/(2)(cot””( beta)/(2) – tan “”(beta)/(2)) tan “”(x)/(2) – tan^(2) (alpha)/(2) =0`
`rArr (tan “”(x)/(2) + cot””(beta)/(2) tan””(alpha)/(2)) (tan””(x)/(2) – tan””(beta)/(2) tan””(alpha)/(2)) =0`
`rArr tan((x)/(2)) = – tan((alpha)/(2)) cot((beta)/(2))`
or `” ” tan ((x)/(2)) = tan((alpha)/(2) tan((beta)/(2))`
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)`
Marlo Parsa
Asked: 2 years ago2022-10-29T05:28:41+05:30
2022-10-29T05:28:41+05:30In: General Awareness
If `cosx-sinalphacotbetasinx=cosa ,`then the value of `tan(x/2)`is`-tan(alpha/2)cot(beta/2)`(b) `tan(alpha/2)tan(beta/2)“-cot((alphabeta)/2)tan(beta/2)`(d) `cot(alpha/2)cot(beta/2)`
A. `-tan (alpha2)cot(beta2)`
B. `tan (alpha//2)tan(beta//2)`
C. `-cot(alpha//2)tan(beta//2)`
D. `cot(alpha//2)cot(beta//2)`
If `cosx-sinalphacotbetasinx=cosa ,`then the value of `tan(x/2)`is`-tan(alpha/2)cot(beta/2)`(b) `tan(alpha/2)tan(beta/2)“-cot((alphabeta)/2)tan(beta/2)`(d) `cot(alpha/2)cot(beta/2)`
A. `-tan (alpha2)cot(beta2)`
B. `tan (alpha//2)tan(beta//2)`
C. `-cot(alpha//2)tan(beta//2)`
D. `cot(alpha//2)cot(beta//2)`
A. `-tan (alpha2)cot(beta2)`
B. `tan (alpha//2)tan(beta//2)`
C. `-cot(alpha//2)tan(beta//2)`
D. `cot(alpha//2)cot(beta//2)`
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`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.