Question

What does `cos(arctan(-1))+sin(arc csc(-1))` equal?

Answer 1

`cos(arctan(-1))+sin(arc csc(-1))=-1+-1/sqrt2`

Explanation:

`cos(arctan(-1))` means `cosalpha` of an angle `alpha`, where `tanalpha=-1`. `tanalpha=-1` for `alpha=(3pi)/4` or `alpha=(-pi)/4`.

`cos((3pi)/4)=-1/sqrt2` and `cos((-pi)/4)=1/sqrt2`. Hence `cos(arc tan(-1))=+-1/sqrt2`

`sin(arc csc(-1))` means `sinbeta` of an angle `beta`, where `cscbeta=-1`.

`cscbeta=-1` for `beta=(3pi)/2` and `sin((3pi)/2)=-1`. Hence `sin(arc csc(-1))=-1`

Hence, `cos(arctan(-1))+sin(arc csc(-1))=-1+-1/sqrt2`