Trigonometric Identity Verification | Simplifying sin(4x) and Solving for x

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The discussion focuses on verifying the trigonometric identity sin(4x) = 8cos^3(x)sin(x) - 4sin(x)cos(x). The user simplifies the right side using double angle identities and applies the Pythagorean identity, leading to a series of transformations. After several simplifications, the user confirms that both sides of the equation are equal, ultimately arriving at the identity through various trigonometric identities. The process demonstrates the effectiveness of working from both sides of the equation to verify the identity. The conclusion is that the identity holds true, confirming the initial equation.
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Homework Statement



sin(4x) = 8cos3(x)sin(x)-4sin(x)cos(x)

Homework Equations



All trigonometric identities

The Attempt at a Solution



I can simplify the right side using the double angle identity to:

sin(4x) = 4sin(2x)cos2(x)-2sin(2x)

However, now I'm not sure what to do. Did I take a step in the wrong direction?
 
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Never mind, I found the solution myself, here is my process. I was on the right track.

sin(4x) = 4sin(2x)cos2(x)-2sin(2x)

Pythagorean Identity:

sin(4x) = 4sin(2x)(1-sin2x)-2sin(2x)

FOIL:

sin(4x) = 4sin(2x)-4sin(2x)sin2(x)-2sin(2x)

Half Angle Identity:

sin(4x) = 4sin(2x)-4sin(2x)[(1-cos(2x)/2]-2sin(2x)

simplify:

sin(4x) = 4sin(2x)-[4sin(2x)+4sin(2x)cos(2x)]/(2)-2sin(2x)

simplify more:

sin(4x) = 4sin(2x)-2sin(2x)+2sin(2x)cos(2x)-2sin(2x)

sin(4x) = 4sin(2x)-4sin(2x)+2sin(2x)cos(2x)

sin(4x) = 2sin(2x)cos(2x)

Double Angle Identity:

sin(4x) = sin(4x)
 
You can save yourself a lot of typing by working from the left side to the right.

sin(4x) = sin(2(2x)) = 2sin(2x)cos(2x)
= 4sin(x)cos(x)(cos2(x) - sin2(x))
= 4sin(x)cos(x)(2cos2(x) - 1)
= 8sin(x)cos3(x) - 4sin(x)cos(x)
QED
 

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