Question

How do you find the exact value of `cos(tan^-1 (5/12)+cot^-1 (5/12))`?

Answer 1

`cos(tan^(−1)(5/12)+cot^(−1)(5/12))=cos(pi/2)=0`

Explanation:

Remember the trigonometric identity

`tantheta=cot(pi/2-theta)=x` i.e.

`tan^-1x=theta` and `cot^-1x=(pi/2-theta)`

Hence, for any `x`, `tan^(−1)x+cot^(−1)x=(theta+pi/2-theta)=pi/2`

Hence for any `x`, `cos(tan^(−1)x+cot^(−1)x)=cos(pi/2)=0`

Note that `x` is equal to `tantheta`, `x` can take any value `{-oo,oo}` (as this is the range of `tantheta`) including `5/12`.