Question

2tan-¹(-1/3)+cos-¹(3/5)=π/2​

Answer 1

2 tan -1 (-1/3) + cos -1 (3/5)

let 2tan -1 (-1/3) = A

tan -1 (-1/3) = A/2

tan (A/2) = -1/3

cos A = (1 - tan^2(A/2))/(1 + tan^2(A/2))

= (1 - (-1/3)^2)/(1 + (-1/3)^2)

= (1 - 1/9) / (1 + 1/9)

= (8/9)/(10/9)

= 4/5

A = cos -1 (4/5)

the problem becomes

cos -1 (4/5) + cos -1 (3/5)

= cos -1 ((4/5)*(3/5) - √(1-(4/5)^2 *√(1 - (3/5)^2

= cos -1 (12/25 - (3/5)*(4/5))

= cos -1 (12/25 - 12/25)

= cos -1 0

= π/2