The function f, defined by )x(f = cos x + sin ,x is an odd function
True /false
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Step-by-step explanation:
Recall that the definition of an even function is
f
(
x
)
=
f
(
−
x
)
and the definition of an odd function is
f
(
x
)
=
−
f
(
x
)
Let's check either of these properties for our function
f
(
x
)
=
cos
(
x
)
⋅
sin
(
x
)
taking into account that
cos
(
x
)
is an even function because
cos
(
x
)
=
cos
(
−
x
)
and
sin
(
x
)
is an odd function because
sin
(
−
x
)
=
−
sin
(
x
)
f
(
−
x
)
=
cos
(
−
x
)
⋅
sin
(
−
x
)
=
=
cos
(
x
)
⋅
[
−
sin
(
x
)
]
=
−
cos
(
x
)
⋅
sin
(
x
)
=
−
f
(
x
)
Therefore,
f
(
x
)
=
cos
(
x
)
⋅
sin
(
x
)
is an odd function.
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