Hardest Identity Evar involving sum and differences

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SUMMARY

The equation sin(x) + sin(3x) + sin(5x) + sin(7x) can be proven to equal 4cos(x)cos(2x)sin(4x) through the application of sum-to-product identities. The key steps involve rewriting the left-hand side using the identities sin(a+b) and sin(a-b) to simplify the expression. This method effectively reduces the complexity of the proof, allowing for the cancellation of terms and leading to the desired equality.

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Homework Statement


sin (x) + sin (3x) + sin (5x) + sin (7x) = 4cos(x)cos(2x)sin(4x)

Homework Equations


sin(a+b)=sin(a)cos(b)+sin(b)cos(a)
sin(a-b)=sin(a)cos(b)-sin(b)cos(a)

The Attempt at a Solution


Me and four of my classmates have tried to do this proof and it kicked our ***.
Please help.
 
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amd123 said:

Homework Statement


sin (x) + sin (3x) + sin (5x) + sin (7x) = 4cos(x)cos(2x)sin(4x)

Homework Equations


sin(a+b)=sin(a)cos(b)+sin(b)cos(a)
sin(a-b)=sin(a)cos(b)-sin(b)cos(a)

The Attempt at a Solution


Me and four of my classmates have tried to do this proof and it kicked our ***.
Please help.

Here's what worked for me.
sin(x) + sin(3x) + sin(5x) + sin(7x) = sin(2x - x) + sin(2x + x) + sin(6x -x) + sin(6x + x)

Expand the terms on the right side, and several terms will drop out. You will need to apply the same trick again.
 

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