Unformatted text preview: we
can get sin(t) in terms of x, then we can write cos(t) = cot(t) sin(t) and be done. The
identity 1 + cot2 (t) = csc2 (t) holds for all t in (0, π ) and relates cot(t) and csc(t) = sin(t) ,
so we substitute cot(t) = 2x and get 1 + (2x)2 = csc2 (t). Thus, csc(t) = ± 4x2 + 1
and since t is between 0 and π , we know csc(t) > 0, so we choose csc(t) = 4x2 + 1.
This gives sin(t) = √4x2 +1 , so that cos(t) = cot(t) sin(t) = √42x+1 . Since arccot(2x) is
deﬁned for all real numbers x and we encountered no additional restrictions on t, we
have the equivalence cos (arccot(2x)) = √42x+1 for all real numbers x.
Why not just start with 1−x 2 and ﬁnd its domain? After all, it gives the correct answer - in this case. There are
lots of incorrect ways to arrive at the correct answer. It pays to be careful. 708 Foundations of Trigonometry The last two functions to invert are secant and cosecant. There are two generally acceptable ways
to restrict the domains of these functions so that they are one-to-one. One approa...
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