How do you simplify #Cos[tan^-1(-1)]#?

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Answer 1 · 2016-06-26T11:40:29

#1/sqrt 2#, for the principal value #-pi/4# of #tan^(-1)(-1)#.
Against , #(3pi)/4#, the answer is #-1/sqrt 2#.

If #a = tan^(-1)(-1), tan a = -1.#,

a is in either 4th quadrant with the principal value #-pi/4# or in

the 2nd, as #(3pi)/4#.

So, the given cosine cos a is

#cos (-pi/4)=1/sqrt 2# or

#cos((3pi)/4)=cos (pi-pi/4)=-cos(pi/4)=-1/sqrt 2#.

In brief, the answer is #+-1/sqrt 2# for the general value of #tan^(-1)(-1)#..