Question

Write the principal value of `tan^(-1)+cos^(-1)(-1/2)`

Answer 1

`Let tan^(-1)=y and cos^(-1)(-1/2)=z`

`tany=1=tan(pi/4) and cosz=-1/2=-cos(pi/3)=cos(pi-pi/3)=cos((2pi)/3)`

The ranges of principal value branch of tan⁻¹ and cos⁻¹ are `(-pi/2,pi/2)and[0,pi] ` respectively

`therefore tan^(-1)=pi/4 and cos^(-1)(-1/2)=2pi/3`

`therefore tan^(-1)(1)+cos^(-1)(-1/2)=pi/4+(2pi)/3=(11pi)/12`