if arctan (3/5) = X, what is arccsc (cotan X)?

Question

evaluate to exact value ( no decimals)

thank you :)

Answer 1

arctan (3/5) = X
3/5 = tan X
cotan X = 5/3

Let Y = arccsc (cotan X) = arccsc (5/3)
csc Y = 5/3
sin Y = 3/5

This is a 3-4-5 triangle where, relative to Y, the opposite side is 3, the adjacent side is 4, and the hypoteneuse is 5. You want Y, since Y = arccsc (cotan X), and Y is the angle whose sine is 3/5. That angle is arcsin (3/5) = 36.87 degrees, but you want an exact answer -- no decimals. In that case, the best you can do is to say

Y = arccsc (cotan X) = arcsin (3/5) (answer)

and let it go at that.