Question

Преобразовать в произведение следующие выражения: $sin\frac{2\pi }{9}+sin\frac{\pi }{9} -2cos\frac{\pi }{18}$
$\frac{\sqrt{2}-cosx-sinx }{sinx-cosx}$
3-4cos4x+cos8x-8cos^4*2x
Ответы:
1.$-cos\frac{\pi }{18}$
2.$tg(\frac{x}{2}-\frac{\pi }{8})$
3.-8cos4x

Answer 1

1.sin2π/9+sinπ/9-2cosπ/18=(sin2π/9+sinπ/9)-2cosπ/18=

2sinπ/6*cosπ/18-2cosπ/18=2*0.5cosπ/18-2cosπ/18=-cosπ/18

2. (√2-cosx-sinx)/(sinx-cosx)=(√2-(sinx+sin(π/2-x)))/(sinx-sin(π/2-x))=

(√2-2sinπ/4*cos((x-π/4)))/(2sin(x-π/4)*cosπ/4)=

√2(1-cos((x-π/4))/(√2*sin(x-π/4))=2sin²(x/2-π/8)/((2sin(x/2-π/8)*(cos(x/2-π/8))=

sin(x/2-π/8)/((cos(x/2-π/8))=tg(x/2-π/8)

3. 3-4cos4x+cos8x-8cos⁴2x= 3-4cos4x+cos²4x-sin²4x-2*(1+cos4x)²=

3-4cos4x+cos²4x-sin²4x-2-4cos4x-cos²4x=3-4cos4x+2cos²4x-1-2-4cos4x

-2cos²4x=-8cos4x