Question

Verify the identity
sin4(t)=4sin(t)cos^3(t)-4sin^3(t)cos(t)

Answer 1

@jim_thompson5910

Answer 2

@Directrix

Answer 3

so this is

\[\Large \sin^4(t)=4\sin(t)\cos^3(t)-4\sin^3(t)\cos(t)\]

right?

or is it

\[\Large \sin(4t)=4\sin(t)\cos^3(t)-4\sin^3(t)\cos(t)\]

Answer 4

its the second one

Answer 5

i bet some factoring could help you

Answer 6

then some use of some double angle identities

Answer 7

would i use this as a factor(a-b)^2

Answer 8

factor out what both terms on the right have in common
which is 4sin(t)cos(t)

Answer 9

like that?

Answer 10

\[4 \sin(t) \cos(t)( \cos^2(t)-\sin^2(t)) \\ 2 \cdot 2 \sin(t) \cos(t) \cdot (\cos^2(t)-\sin^2(t))\]

Answer 11

do you know what 2sin(t)cos(t) and cos^2(t)-sin^2(t) can be replaced with?

Answer 12

sin2(t)

Answer 13

then cos2(t)

Answer 14

and i think you mean this
sin(2t)=2sin(t)cos(t)
and
cos(2t)=cos^2(t)-sin^2(t)
and if so yes

Answer 15

\[4 \sin(t) \cos(t)( \cos^2(t)-\sin^2(t)) \\ 2 \cdot 2 \sin(t) \cos(t) \cdot (\cos^2(t)-\sin^2(t)) \\ 2 \sin(2t) \cos(2t)\]

Answer 16

yes i do mean that

Answer 17

you have one final thing to realize

Answer 18

which is?

Answer 19

well recall sin(2u)=2sin(u)cos(u)

Answer 20

so if u=2x
then replacing u with 2x gives us
sin(2*2x)=2sin(2x)cos(2x) correct?