Let `alpha,betaa n dgamma` be some angles in the first quadrant satisfying `tan(alpha+beta)=(15)/8a n dcos e cgamma=(17)/8,` then which of the following hold(s) good? (a) `alpha+beta+gamma=pi` (b)`cotalpha+cotbeta+cotgamma=cotalphacotbetacotgamma` (c)`tanalpha+tanbeta+tangamma=tanalphatanbetatangamma` (d)`tanalphatanbeta+tanbetatangamma+tangammatanalpha=1`

Options

A. `alpha+beta+gamma=pi`
B. `cot alpha cot beta cot gamma=cot alpha+cot beta+cotgamma`
C. `tan alpha+tan beta+beta gamma=tan alpha tan beta tan gamma`
D. `tan alpha tan beta+tan beta tan gamma+tan gamma tan alpha=1`

Answer

B, D

Explanation

`tan (alpha+beta)=(15)/(8)`
and `co sec gamma=(17)/(8)`
`rArr tan gamma=(8)/(15)`
`therefore alpha+beta+gamma=(pi)/(2)`
Hence, (2) is true.
Also `tan(alpha+beta)=(tan alpha+tan beta)/(1-tan alpha tan beta)`
or `cot gamma=(tan alpha+tan beta)/(1-tan alpha tan beta)`
or `tan alpha tan beta+tan beta tan gamma+tan gamma tan alpha=1`.