If `P=x+1/x` and `P(tan alpha)=2, P(tan beta)=2sqrt2, P(cot gamma)=4` where `alpha, beta, gamma` are acute angles and sum of all possible distinct values of `alpha, beta` and `gamma `is given by `(mpi)/n` where `m` and `n` are coprime, then find the value of `(m+n)`

Explanation

`tanalpha+1/tanalpha=2impliestan alpha=1impliesa (pi)/(4)" "("only acute solution")`
`tan alpha+(1)/(tan alpha)=2sqrt2impliestanbeta=sqrt2-1,sqrt2+1" " beta=(pi)/(8),(3pi)/(8)`
`cotgamma+(1)/(cotgamma)=4impliescot gamma=2+sqrt3, 2-sqrt3" "gamma=(pi)/(12),(5pi)/(12)`
sum of all values `=(5pi)/(4)=(mpi)/(n)`
`thereforem=5and n=4 Ans."]"`