Prove the identity 4 \sin(x) \cos(x) = ( \sin(4x) )/( \cos(2x) ) ​

Question

Prove the identity

`4 \sin(x) \cos(x) = ( \sin(4x) )/( \cos(2x) )`
​

Answer 1

Below

Explanation:

● 4 sin(x) cos(x) = sin(4x)/cos(2x)

Let's prove that:

● 4 sin(x) cos(x) cos(2x) = sin(4x)

It's easier to prove it than the first one

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● 4 sin(x) cos(x) cos (2x)

● [2 sin(x) cos(x)] 2 cos(2x)

We khow that [2 sin(x) cos(x)]= sin(2x)

So:

● sin(2x) 2 cos(2x)

Based on the same relation

● sin(4x)

It's proved

Answer 2

=> R.H.S

`( \sin(4x) )/( \cos(2x) ) = ( \sin(2x + 2x) )/( \cos(2x) )`

`= ( 2 \sin(2x) \cos(2x) )/( \cos(2x) )`

`= 2 \sin(2x)`

`= 2(2 \sin(x) \cos(x) )`

`= 4 \sin(x) \cos(x)`

R.H.S = L.H.S

PROVED!