Question

prove that sin 3x=3 sin x-4 sin 3x​

Answer 1

Answer: -

LHS↓

$\implies \sin \: 3x \\ \\ \implies \sin \:( 2x + x) \\ \\ \implies \sin \: 2x. \cos \: x + \cos \:2 x. \sin \: x \\ \\ \implies 2 \sin \: x \cos \: x. \cos \: x + (1 - 2 \sin \: {}^{2} x) \sin \: x \\ \\ \implies \: 2 \sin \: x. \cos {}^{2} x + \sin \: x - 2 \sin {}^{3} x \\ \\ \implies \: 2 \sin \: x(1 - { \sin}^{2} x) + \sin \: x - 2 \sin \: {}^{3} x \\ \\ \implies \: 2 \sin \: x - 2 \sin {}^{3} x + \sin \: x - 2 { \sin}^{3} x \\ \\ \implies \: 3 \sin \: x - \: 4 \sin {}^{3} x$

hence proved

_______________________

formula use:↓

$\implies \pink{ \sin \: (a + b) = \sin \: a \cos \: b + \cos \: a \sin \: b} \\ \\ \implies \pink{ \cos \: 2x = 1 - 2 \sin {}^{2} x}$

Answer 2

Step-by-step explanation:

sin3x=sin ( 2x + x )

=sin2xcosx + cos 2x sin x

=2sinx cosx cosx + ( 1 − 2 sin²x )sinx

=2sinx( 1 − sin²x ) + sinx − 2sin 3x

=3sinx − 4sin 3x

I'm good