2cos ( x+ П/4) cos( 2x + П/4) + sin3x

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вопрос

To expand the expression, we will first expand the product of the two cosine terms using the double angle identities:

cos(2x) = cos^2(x) - sin^2(x)

Therefore, 2cos(x + П/4)cos(2x + П/4) = 2(cosxcos2x - sinxsin2x)

= 2(cosx(cos^2(x) - sin^2(x)) - sinx2cosx*sinx)

= 2(cosxcos^2(x) - cosxsin^2(x) - 2sin^2(x)cosx)

= 2(cos^3(x) - cosx*sin^2(x) - 2sin^2(x)cosx)

Next, we will expand the sine term sin3x using the double angle identity:

sin3x = 3sinx - 4sin^3(x)

Therefore, the expanded expression becomes:

2cos(x)(cos^2(x) - sin^2(x)) - cos(x)sin^2(x) - 2sin^2(x)cos(x) + 3sin(x) - 4*sin^3(x)

= 2cos(x)cos^2(x) - 2cos(x)sin^2(x) - cos(x)sin^2(x) - 2sin^2(x)cos(x) + 3sin(x) - 4*sin^3(x)

= 2cos(x)cos^2(x) - 3cos(x)sin^2(x) - 2sin^2(x)cos(x) + 3sin(x) - 4sin^3(x)