MATH 1310: CSM
Tutorial: the tangent and arctangent functions
For the purposes of this tutorial, you may want to recall that
d
sin x = cos x,
dx
d
cos x =
dx
sin x,
tan x =
sin x
,
cos x
sec x =
1
.
cos x
1. Use the above definitions of tan x and sec x, together with the quotient rule
✓
◆
d f (x)
g(x)f 0 (x) f (x)g 0 (x)
=
,
dx g(x)
(g(x))2
and the trig identity cos2 x + sin2 x = 1, to show that
d
tan x = sec2 x.
dx
d
d
cos x sin x sin x cos x
d
cos x cos x sin x( sin x)
dx
dx
tan x =
=
dx
(cos x)2
(cos x)2
=
(cos x)2 + (sin x)2
1
=
= sec2 x.
(cos x)2
(cos x)2
ANSWERS
2. On the axes below is a graph of the function y = f (x) = tan x, for
⇡
2
<x<
⇡
2.
y
2
1
! Π2
(As x approaches
! Π4
!1
Π
4
⇡/2 from the right, the graph of f (x) shoots o↵ towards
from the left, the graph of f (x) shoots o↵ towards 1.)
Π
2
x
1; as x approaches ⇡/2
(a) Explain why this function f (x), when restricted to the given domain, has an inverse function
g(x) = arctan x. (Pronounced “arctangent of x.”)
The above function y = tan(x) is steadily increasing, so there is only one input for each output,
so that, if we interchange the roles of x and y, we get only one output per input . This is what it
means to say y = tan(x) has an inverse function, which we denote by g(x) = arctan(x).
(b) Sketch the graph of y = g(x) = arctan x, on the axes below.
Π
2
y
Π
4
!1
! Π4
! Π2
1 2
x
3. (a) Note that, because we have defined arctan x as the inverse of tan x, we have
tan(arctan x) = x.
Di↵erentiate both sides of this equation to find a formula for the derivative of arctan x. Express
your answer in terms of sec(arctan x). (You’ll need the result of problem 1 here.)
Di↵erentiating the equation
tan(arctan x) = x
using the chain rule gives
sec2 (arctan) ·
d
arctan x = 1,
dx
or, solving,
d
1
arctan x =
.
2
dx
sec (arctan x)
(b) It’s a fact that, for any number ✓ between
⇡/2 and ⇡/2,
sec2 ✓ = 1 + tan2 ✓.
Use this fact, and the result of problem 6(a) above, to show that
d
1
arctan x =
.
dx
1 + x2
d
1
1
1
1
arctan x =
=
=
=
,
2
2
2
dx
sec (arctan x)
[sec(arctan x)]
[1 + tan(arctan x)]
1 + x2
the last step because, again, tan(arctan x) = x.