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5.2 when we saw (−2)2 = 2 as opposed to −2. Additionally, even though the expression 1 − 2x2 is
deﬁned for all real numbers, the equivalence cos (2 arcsin(x)) = 1 − 2x2 is valid for only −1 ≤ x ≤ 1.
This is akin to the fact that while the expression x is deﬁned for all real numbers, the equivalence
√2
( x) = x is valid only for x ≥ 0. 3 In other words, the angle θ = t radians is a Quadrant I or II angle where sine is nonnegative.
Alternatively, we could use the identity: 1 + tan2 (t) = sec2 (t). Since we are given x = cos(t), we know sec(t) =
1
1
= x . The reader is invited to work through this approach to see what, if any, diﬃculties arise.
cos(t)
4 10.6 The Inverse Trigonometric Functions 705 The next pair of functions we wish to discuss are the inverses of tangent and cotangent. First, we
restrict f (x) = tan(x) to its fundamental cycle on − π , π to obtain f −1 (x) = arctan(x). Among
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other things, note that the vertical asymptotes x = − π and x = π of the graph of f (x) = tan(x)
2
2
become the horizontal asymptotes y = − π and y = π of the graph of f −1 (x) = arctan(x).
2
2 y y 1 π
2 −π −π
2
4 π
4 π
2 x π
4 −1
−1 x 1
−π
4 reﬂect across y = x f (x) = tan(x), − π < x <
2 −− − − − −→
−−−−−− π
.
2 switch x and y coordinates −π
2
f −1 (x) = arctan(x). Next, we restrict g (x) = cot(x) to its fundamental cycle on (0, π ) to ob...

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