Show that `2tan^-1 (-3)=-pi/2+tan^-1(-4/3)`

Answer

N/a

Explanation

`LHS = 2tan^(-1)(-3)=-2tan^(-1)3 , [:' tan^(-1)(-x)=-tan^(-1)x,x in R]`
` = -[cos^(-1)'(1-3^(2))/(1+3^(2))], [:' 2tan^(-1)x = cos^(-1)'(1-x^(2))/(1+x^(2)), x ge0]`
`= - [cos^(-1)'((-8)/(10))] = - [cos^(-1)((-4)/(5))]`
`= -[pi-cos^(-1)(4/5)] , { :' cos^(-1)(-x) = pi- cos^(-1)x, x in [-1,1]}`
`= - pi + cos^(-1)(4/5) , ["let" cos^(-1) (4/5) = theta = cos theta = 4/5 rArr tan theta = 3/4 rArr theta = tan^(-1)'3/4]`
`= -pi + tan^(-1)(3/4) = -pi+[pi/2-cot^(-1)(3/4)]`
`= - (pi)/(2)-cot^(-1)'3/4 = - pi/2 - tan^(-)'4/3`
`= - (pi)/(2) + tan^(-1)'((-4)/(3)) , [:' tan^(-1)(-x) = - tan^(-1)x]`
`= RHS`