sin (arctan sq.rt.3) + cos (arcctn sq.rt. 3)?

Question

evaluate (as exact value- no decimals)

thank you so much

Answer 1

sin(arctan(sqrt(3))) + cos(arctan(sqrt(3)))

First off, arctan(sqrt(3)) is like answering the question

tan(x) = sqrt(3). Since arctan is such that the restriction is from -pi/2 to pi/2,

x = pi/3

Therefore, our question becomes

sin(pi/3) + cos(pi/3) = sqrt(3)/2 + 1/2
= [sqrt(3) + 1]/2

Answer 2

tan(π/3) = √3 so arctan(√3) = π/3.
cot(π/6) = √3 so arccot(√3) = π/6.
___________

sin(arctan(√3)) + cos(arccot(√3))
= sin(π/3) + cos(π/6) = √3/2 + √3/2 = √3