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Product-to-sum Identity Example 2

Q : Which is 2*sin(4x)*cos(2x) written as a sum containing only sines?

A. sin 6x – sin 2x

B. sin 5x – sin3x

C. sin 5x + sin 3x

D. sin 6x + sin 2x

A:  This is another product-to-sum identity, only it is an the identity that involves sines and cosines mixed.  The correct identity is:

sin(a)cos(b) = [sin(a + b) + sin(a – b)]/2

Multiply the 2 across to get:

2*sin(a)cos(b) = sin(a + b) + sin(a – b)

See how this now looks like our problem?

2*sin(4x)*cos(2x)!!

So, our a = 4x and our b = 2x… Plug those in to get:

2*sin(4x)*cos(2x) = sin(4x + 2x) + sin(4x – 2x)

2*sin(4x)*cos(2x) = sin(6x) + sin(2x)

This is answer D

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