Tan [ cos^ -1 (1/√3) + sin^ -1(1/√3)]

we know cos ^- 1 (x) = pi/2 - sin ^-1 (x) ( you can find it from rt angle triangle)

or from cos t = sin (90-t) and taking arc cos

so cos ^- 1 (x) +- sin ^-1 (x) = pi/ 2

so cos^ -1 (1/√3) + sin^ -1(1/√3) = pi/ 2

and hence tan [ cos^ -1 (1/√3) + sin^ -1(1/√3)] = tan pi/2 = infinity
Note:
actually it is infinite(or to be more precise tends to be infinite) from left and - infinite for right so strictly speaking the value does not exist.