If `2sin^(-1)((9-4x^(2))/(9+4x^(2)))+3cos^(-1)((12x)/(9+4x^(2)))+tan^(-1)((12x)/(9-4x^(2)))=(lambda pi)/(2)` for `x=(-17)/(13)` then `lambda`

Answer

5

Explanation

`Put `(2x)/(3)=tan theta` then
`2sin^(-1)(cos2theta)+3cos^(-1)(sin2theta)+tan^(-1)tan2theta=lambda.(pi)/(2)`
`implies 2((pi)/(2)-cos^(-1)cos2theta)+3((pi)/(2)-sin^(-1)sin2theta)+tan^(-1)tan 2 theta =(lambdapi)/(2)`
for `x=(-17)/(13)implies(-pi)/(2)lt2thetalt0`
`therefore 2((pi)/(2)+2theta)+3((pi)/(2)-2theta)+2theta=(lambdapi)/(2)`
`(5pi)/(2)=(lambdapi)/(2)implieslambda=5`