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If alpha+beta=pi/2 and beta+gamma=alpha ...

If `alpha+beta=pi/2` and `beta+gamma=alpha` then `tanalpha` equals

A

`2(tan beta + tan gamma)`

B

`tan beta + tangamma`

C

`tanbeta + 2tan gamma`

D

`2tan beta + tan gamma`

Text Solution

Verified by Experts

The correct Answer is:
c
Given, `alpha+beta=pi//2`
`implies alpha=(pi//2)-beta`
`implies tan alpha = tan (pi//2-beta)`
`implies tan alpha = cot beta`
`implies tan alpha tan beta = 1`
Again, `beta + gamma = alpha " [given]"`
`implies gamma = (alpha - beta)`
`implies tan gamma = tan(alpha - beta)`
`implies tan gamma = (tanalpha-tanbeta)/(1+tanalphatanbeta)`
`implies tan gamma = (tan alpha-tanbeta)/(1+1)`
`therefore 2 tan gamma = tan alpha - tan beta`
`implies tan alpha=tanbeta + 2 tan gamma`
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