Question

Prove that cosh3z equals to 4cosh^3z

Answer 1

Since the angle sum formula of sinus:
sin
(
α
+
β
)
=
sin
α
cos
β
+
cos
α
sin
β
,
and the double angle formula of cosine:
cos
(
2
α
)
=
cos
2
α
−
sin
2
α
=
2
cos
2
α
−
1
=
1
−
2
sin
2
α
then:
sin
(
3
x
)
=
sin
(
2
x
+
x
)
=
sin
(
2
x
)
cos
x
+
cos
(
2
x
)
sin
x
=
=
(
2
sin
x
cos
x
)
⋅
cos
x
+
(
1
−
2
sin
2
x
)
sin
x
=
=
2
sin
x
cos
2
x
+
sin
x
−
2
sin
3
x
=
=
2
sin
x
(
1
−
sin
2
x
)
+
sin
x
−
2
sin
3
x
=
=
2
sin
x
−
2
sin
3
x
+
sin
x
−
2
sin
3
x
=
=
3
sin
x
−
4
sin
3
x
.

Answer 2

Since the angle sum formula of sinus:

sin(α+β)=sinαcosβ+cosαsinβ,

and the double angle formula of cosine:

cos(2α)=cos2α−sin2α=2cos2α−1=1−2sin2α

then:

sin(3x)=sin(2x+x)=sin(2x)cosx+cos(2x)sinx=

=(2sinxcosx)⋅cosx+(1−2sin2x)sinx=

=2sinxcos2x+sinx−2sin3x=

=2sinx(1−sin2x)+sinx−2sin3x=

=2sinx−2sin3x+sinx−2sin3x=

=3sinx−4sin3x.

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