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Tangent/Cotangent/Derivative/Fact
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The
tangent function
R
∖
(
π
2
+
Z
π
)
⟶
R
,
x
⟼
tan
x
,
{\displaystyle \mathbb {R} \setminus {\left({\frac {\pi }{2}}+\mathbb {Z} \pi \right)}\longrightarrow \mathbb {R} ,x\longmapsto \tan x,}
is
differentiable
, with
tan
′
(
x
)
=
1
cos
2
x
,
{\displaystyle {}\tan \!'(x)={\frac {1}{\cos ^{2}x}}\,,}
and the
cotangent function
R
∖
Z
π
⟶
R
,
x
⟼
cot
x
,
{\displaystyle \mathbb {R} \setminus \mathbb {Z} \pi \longrightarrow \mathbb {R} ,x\longmapsto \cot x,}
is differentiable, with
cot
′
(
x
)
=
−
1
sin
2
x
.
{\displaystyle {}\cot \!'(x)=-{\frac {1}{\sin ^{2}x}}\,.}
Proof
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