Question

Thu gọn các biểu thức sau:

a. \(sin^6a+c\text{os}^6a+3sin^2a.c\text{os}^2a\)

b.\(sin^4a-c\text{os}^4a-\left(sina+c\text{os}a\right)\left(sina-c\text{os}a\right)\)

c.\(c\text{os}^2a+tan^2a.c\text{os}^2a\)

d.\(c\text{os}^2a+tan^2a.c\text{os}^2a\)

Answer 1

a) \(sin^6x+cos^6x+3sin^2x.cos^2x\)

\(=\left(sin^2x+cos^2x\right)\left(sin^4x-sin^2x.cox^2x+cos^4x\right)+3sin^2x.cos^2x\)

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

\(=sin^4x+2sin^2x.cox^2x+cos^4x=\left(sin^2x+cos^2x\right)^2=1\text{​​}\text{​}\)

b) \(sin^4x-cos^4x-\left(sinx+cosx\right)\left(sinx-cosx\right)\)

\(=\left(sin^2x+cos^2x\right)\left(sin^2x-cos^2x\right)-\left(sin^2x-cos^2x\right)\)

\(=1\left(sin^2x-cos^2x\right)-\left(sin^2x-cos^2x\right)=0\)

c) \(cos^2x+tan^2x.cos^2x\)

\(=cos^2x+\dfrac{sin^2x}{cos^2x}.cos^2x=sin^2x+cos^2x=1\)