Question

Упростить: $2cos(a+b)cos(a-b)-1-2sin^2b$

Answer 1

2cos(α+β)*cos(α-β)-1-2sin²β=

2*0.5(cos(α+β+α-β)+cos(α+β-α+β))-1-2sin²β=

cos2α+cos2β-1-2sin²β=2cos²α- 1+2cos²β-1-1-2sin²β=

2cos²α-1+2-2sin²β-2-2sin²β=2cos²α-1-2sin²β-2sin²β=-(4sin²β-2cos²α+1)

Answer 2

$2cos(a+b)cos(a-b)-1-2sin^2b=2\cdot \dfrac{1}{2}\cdot \Big(cos2a+cos2b\Big)-1-2sin^2b=\\\\=cos2a+cos2b-1-2sin^2b=\\\\=(cos^2a-sin^2a)+(cos^2b-sin^2b)-1-2sin^2b=\\\\=(cos^2a-(1-cos^2a))+(cos^2b-(1-cos^2b))-1-2sin^2b=\\\\=(2cos^2a-1)+(2cos^2b-1)-1-2(1-cos^2b)=\\\\=2cos^2a-1+2cos^2b-1-1-2+2cos^2b=2cos^2a+4cos^2b-5$