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The value of expression of (alphatangamm...

The value of expression of `(alphatangamma+betacotgamma)(alphacotgamma+betatangamma)-4alphabetacot^2 2gamma` depends on `alpha` (b) `beta` (c) `gamma` (d) none of these

A

dependent on `alpha`

B

independent of `gamma`

C

dependent on `beta`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
A, B, C
`(alpha tan gamma + beta cot gamma) (alpha cot gamma + beta tan gamma)-4alpha bet cot^(2)2gamma`
`=alpha^(2)+(tan^(2)gamma + cot^(2)gamma)alpha betas + beta^(2)-4 alpha beta (cos^(2)2gamma)/(sin^(2)2gamma)`
`= alpha^(2)+beta^(2)+alpha beta[((sin^(2)gamma)/(cos^(2)gamma)+(cos^(2)gamma)/(sin^(2)gamma))-(4(cos^(2)gamma-sin^(2)gamma)^(2))/(4sin^(2)gamma cos^(2)gamma)]`
`=alpha^(2)+beta^(2)+alpha beta [((sin^(4)gamma+cos^(4)gamma)-(cos^(4)gamma +sin^(4)gamma-2 sin^(2)gamma cos^(2)gamma))/(sin^(2)gamma cos^(2)gamma)]`
`=alpha^(2)+beta^(2)+2alpha beta =(alpha+beta)^(2)`, which is independent on `gamma` and dependent of `alpha, beta`.
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