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Question

Evaluate: (cosxcos3x)(sin8x+sin2x)(sin5xsinx)(cos4xcos6x)

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Solution

Given:

(cosxcos3x)(sin8x+sin2x)(sin5xsinx)(cos4xcos6x)

Use the formula:

sina+sinb=2sin(a+b2)cos(ab2)

sinasinb=2cos(a+b2)sin(ab2)

cosacosb=2sin(a+b2)sin(ab2)

(cosxcos3x)(sin8x+sin2x)(sin5xsinx)(cos4xcos6x)=(2sin4x2sin2x2)(2sin10x2cos6x2)(2cos6x2sin4x2)(2sin10x2sin2x2)

=(2sin2xsinx)(2sin5xcos3x)(2cos3xsin2x)(2sin5xsinx)

=1

Thus, the value is 1.


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