Question

Prove this identity, please with steps:
sin5x=sinx(cos^2 2x-sin^2 2x) + 2cosxcos2xsin2x

Answer 1

It's easier to start from the right side, i.e. sinx(cos^2 2x-sin^2 2x) + 2cosxcos2xsin2x.
Now, consider 2cosxcos2xsin2x,
2cosxcos2xsin2x
= cosx(2cos2xsin2x)
Can you use the identity \(\sin(2x) = 2\sin(x)\cos(x)\) to simplify the expression?

Answer 2

cosx(sin(4x))?

Answer 3

Yes, so now, we have \(\sin x(\cos^2( 2x)-\sin^2( 2x)) + \cos(x) \sin(4x)\)

For the first term, \(\sin x(\cos^2(2x)-\sin^2(2x))\)
Can you use the identity \(\cos(2x) = \cos^2(x) -\sin^2(x)\) to simplify the expression?

Answer 4

sinx(cos^2 2x - sin^2 2x) = sinx(cos(4x))

Answer 5

LS= sinxcos4x + cosxsin4x

Answer 6

RS= sinx(cos(4x)) + cosx(sin4x))

Answer 7

RS= sinx cos(4x) + cosx sin(4x) is right.
Now, use the identity \(\sin(a+b)=\sin(a)\cos(b) + \cos(a)\sin(b)\) to simplify the right side, what do you get?

Answer 8

sin(5x)

Answer 9

Yes! Proven! THanks so much!

Answer 10

You're welcome! :D