Question

tan^-1(1/3 tan x/2)​

Answer 1

Answer:

tan^-1{(1/√3)tan(x/2)}

=(1/2) 2tan^-1{(1/√3)tan(x/2)}

=(1/2)cos^-1[{(1-(1/3)tan²(x/2)}/{(1+(1/3)tan²(x/2)}]

=(1/2)cos^-1[{3-tan²(x/2)}/{3+tan²(x/2)}]

=(1/2)cos^-1[{3-(1-cosx)/(1+cosx)}/{3+(1-cosx)/(1+cosx)}]

=(1/2)cos^-1{(2+4cosx)/(4+2cosx)}

=(1/2)cos^-1{(1+2cosx)/(2+cosx)}(Proved)