Question

Rút gọn biểu thức: 3(sin⁸x - cos⁸x) + 4(cos⁶x - 2sin6x) + 6sin⁴x 

 

Answer 1

M = 3(sin^8x-cos^8x) + 4(cos^6x-2sin^6x)+6sin^4x
Ta có:
sin^8(x) - cos^8(x) = [sin^4(x) ]² - [cos^4(x)]²
= (sin²x + cos²x)(sin²x -cos²x).[ sin^4(x) + cos^4(x) ]
= (sin²x -cos²x)[ sin^4(x) + cos^4(x) ]
= sin^6(x) - cos^6(x) + sin²x.cos^4(x) -cos²x.sin^4(x)
Lúc đó M viết lại là:
M = 3.[sin^6(x) - cos^6(x) + sin²x.cos^4(x) -cos²x.sin^4(x) ] + 4.[ cos^6(x) -2sin^6(x) ] + 6sin^4(x)
M = -5sin^6(x) + cos^6(x) -3sin^4(x).cos²x + 3sin²x.cos^4(x) +6sin^4(x)
M = -3sin^(6)x - 3cos²x.sin^4(x) + cos^4(x).sin²x + cos^6(x) - 2sin^6(x) + 2sin²x.cos^4(x) + 6sin^4(x)
M = -3sin^4(x).(sin²x + cos²x ) + cos^4(x).[sin²x + cos²x ] -2sin²x.[sin^4(x) - cos^4(x) ] + 6sin^4(x)
M = 3sin^4(x) + cos^4(x) -2sin²x.[sin²x - cos²x]
M = 3sin^4(x) + cos^4(x) -2sin^4(x) + 2sin²x.cos²x
M = sin^4(x) + 2sin²x.cos²x + cos^4(x)
M = [sin²x + cos²x ]² = 1