\(C=8\left(cos^8x-sin^8x\right)-cos6x-7cos2x\)

\(=8\left[\left(cos^2x\right)^4-\left(sin^2x\right)^4\right]-cos6x-7cos2x\)

\(=8\left[\left(cos^2x\right)^2+\left(sin^2x\right)^2\right]\left[\left(cos^2x\right)^2-\left(sin^2x\right)^2\right]-cos6x-7cos2x\)

\(=8\left[1\right]\left[cos^4x-sin^4x\right]-cos6x-7cos2x\)

\(=8\left[cos^4x-\left(1-cos^2x\right)^2\right]-cos6x-cos2x\)

\(=8\left[cos^4x-1+2cos^2x-cos^4x\right]-cos6x-7cos2x\)

\(=16cos^2x-8-cos6x-7cos2x\)

\(=16cos^2x-8-\left(1-2sin^23x\right)-\left(2cos^2x-1\right)\)

\(=18cos^2x-2sin^23x-1\)

\(=18cos^2x-2\left(1-cos^26x\right)-1\)

\(=20cos^2x-2cos^26x-3\)

Mẫu số là \(-3cos2a\) hay \(-2cos2a\) vậy bạn? -3 không hợp lý

\(\left(sin^4x+cos^4x+cos^2x.sin^2x\right)^2-sin^8x\)

\(=\left(sin^4x+cos^2x\left(cos^2x+sin^2x\right)\right)^2-sin^8x\)

\(=\left(sin^4x+cos^2x\right)^2-sin^8x=\left(sin^4x+cos^2x-sin^4x\right)\left(sin^4x+cos^2x+sin^4x\right)\)

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

Tương tự: \(\left(sin^4x+cos^4x+sin^2xcos^2x\right)^2-cos^8x\)

\(=\left(cos^4x+sin^2x\left(sin^2x+cos^2x\right)\right)^2-cos^8x\)

\(=\left(cos^4x+sin^2x\right)^2-cos^8x\)

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

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

\(\Rightarrow M=2sin^2x.cos^4x+2sin^2x.cos^2x+sin^2x+cos^4x\)

\(M=2sin^2x.cos^2x\left(cos^2x+sin^2x\right)+sin^4x+cos^4x\)

\(M=2sin^2x.cos^2x+sin^4x+cos^4x\)

\(M=\left(sin^2x+cos^2x\right)^2=1\)

a: \(A=2\left(\sin^6x+cos^6x\right)-3\cdot\left(\sin^4x+cos^4x\right)\)

\(=2\cdot\left\lbrack\left(\sin^2x+cos^2x\right)^3-3\cdot\sin^2x\cdot cos^2x\cdot\left(\sin^2x+cos^2x\right)\right\rbrack-3\cdot\left\lbrack\left(sin^2x+cos^2x\right)^2-2\cdot\sin^2x\cdot cos^2x\right\rbrack\)

\(=2\left\lbrack1-3\cdot sin^2x\cdot cos^2x\right\rbrack-3\cdot\left\lbrack1-2\cdot\sin^2x\cdot cos^2x\right\rbrack\)

\(=2-6\cdot\sin^2x\cdot cos^2x-3+6\cdot\sin^2x\cdot cos^2x\)

=2-3

=-1

c: \(C=\frac{\sin^2x}{1+\cot x}+\frac{cos^2x}{1+\tan x}+\sin x\cdot cosx\)

\(=\frac{\sin^2x}{1+\frac{cosx}{\sin x}}+\frac{cos^2x}{1+\frac{\sin x}{cosx}}+\sin x\cdot cosx=\sin^2x:\frac{\sin x+cosx}{\sin x}+cos^2x:\frac{\sin x+cosx}{cosx}+\sin x\cdot cosx\)

\(=\frac{\sin^3x+cos^3x}{\sin x+cosx}+\sin x\cdot cosx\)

\(=\frac{\left(\sin x+cosx\right)\left(\sin^2x-\sin x\cdot cosx+cos^2x\right)}{\sin x+cosx}+\sin x\cdot cosx\)

\(=\sin^2x-\sin x\cdot cosx+cos^2x+\sin x\cdot cosx\)

\(=\sin^2x+cos^2x=1\)

d: \(D=\frac{\cot^2x-cos^2x}{cot^2x}+\frac{\sin x\cdot cosx}{\cot x}\)

\(=\left(\frac{cos^2x}{\sin^2x}-cos^2x\right):\frac{cos^2x}{sin^2x}+\frac{\sin x\cdot cosx}{\frac{cosx}{\sin x}}\)

\(=cos^2x\left(\frac{1}{\sin^2x}-1\right)\cdot\frac{\sin^2x}{cos^2x}+\frac{\sin x\cdot cosx\cdot\sin x}{cosx}\)

\(=\frac{1-\sin^2x}{\sin^2x}\cdot\sin^2x+\sin^2x=1-\sin^2x+\sin^2x=1\)

\(A=\sqrt{\left(1-cos^2x\right)^2+4cos^2x}+\sqrt{\left(1-sin^2x\right)^2+4sin^2x}\)

\(=\sqrt{cos^4x+2cos^2x+1}+\sqrt{sin^4x+2sin^2x+1}\)

\(=\sqrt{\left(cos^2x+1\right)^2}+\sqrt{\left(sin^2x+1\right)^2}\)

\(=sin^2x+cos^2x+2=3\)

b/

\(3\left(sin^8x-cos^8x\right)=3\left(sin^4x+cos^4x\right)\left(sin^4x-cos^4x\right)\)

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

\(=3sin^6x-3sin^4x.cos^2x+3sin^2x.cos^4x-3cos^6x\)

\(\Rightarrow B=-5sin^6x-3sin^4x.cos^2x+3sin^2x.cos^4x+cos^6x+6sin^4x\)

\(=-5sin^6x-3sin^4x\left(1-sin^2x\right)+3cos^4x\left(1-cos^2x\right)+cos^6x+6sin^4x\)

\(=-2sin^6x-2cos^6x+3sin^4x+3cos^4x\)

\(=-2\left(1-3sin^2x.cos^2x\right)+3\left(1-2sin^2x.cos^2x\right)\)

\(=-2+3=1\)

\(=3\left(sin^4x+cos^4x\right)\left(sin^2x-cos^2x\right)+4cos^6x-8sin^6x+6sin^4x\)

\(=3\left(sin^4x+cos^4x\right)\left(sin^2x-cos^2x\right)+4cos^6x-2sin^6x+6sin^4x\left(1-sin^2x\right)\)

\(=sin^6x+3sin^4x.cos^2x+3cos^2x.sin^4x+cos^6x\)

\(=\left(sin^2x+cos^2x\right)^3=1\)