a)
\(A=\left(sin\alpha+cos\alpha\right)^2+\left(sin\alpha-cos\alpha\right)^2\)
\(=1+2sin\alpha cos\alpha+1-2sin\alpha cos\alpha=2\) (không phụ thuộc vào \(\alpha\)).
b)
\(B=sin^4\alpha-cos^4\alpha-2sin^2\alpha+1\)
\(=\left(sin^2\alpha+cos^2\alpha\right)\left(sin^2\alpha-cos^2\alpha\right)-2sin^2\alpha+1\)
\(=sin^2\alpha-cos^2\alpha-2sin^2\alpha+1\)
\(=-sin^2\alpha-cos^2\alpha+1\)
\(=-\left(sin^2\alpha+cos^2\alpha\right)+1=-1+1=0\).

Ta có: \(\sin^2\alpha+\cos^2\alpha=1\forall\alpha\)

\(\Rightarrow\left(\sin^2\alpha+\cos^2\alpha\right)^3=1\Rightarrow\sin^6\alpha+\cos^6\alpha+3\sin^2\alpha\cdot\cos^2\alpha\cdot\left(\sin^2\alpha+\cos^2\alpha\right)=1.\)

\(\Rightarrow E=\sin^6\alpha+\cos^6\alpha+3\sin^2\alpha\cdot\cos^2\alpha=1.\)không phụ thuộc vào \(\alpha\)

Kết quả:

A=1    B=2   C=-4

\(A=\sin^6\alpha+cos^6\alpha+3\sin^2\alpha\cos^2\alpha\left(\sin^2\alpha+\cos^2\alpha\right).\)vì\(\sin^2\alpha+\cos^2\alpha=1\)

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

\(B=2\left(\cos^2\alpha+\sin^2\alpha\right)=2.1=2\)

\(C=\frac{-4\cos\alpha\sin\alpha}{\sin\alpha\cos\alpha}=-4\)

a/ \(A=\frac{cot^2a-cos^2a}{cot^2a}-\frac{sina.cosa}{cota}\)

\(=\frac{\frac{cos^2a}{sin^2a}-cos^2a}{\frac{cos^2a}{sin^2a}}-\frac{sina.cosa}{\frac{cosa}{sina}}\)

\(=\left(1-sin^2a\right)-sin^2a=1\)

b/ \(B=\left(cosa-sina\right)^2+\left(cosa+sina\right)^2+cos^4a-sin^4a-2cos^2a\)

\(=cos^2a-2cosa.sina+sin^2a+cos^2a+2cosa.sina+sin^2a+\left(cos^2a+sin^2a\right)\left(cos^2a-sin^2a\right)-2cos^2a\)

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

\(=2-sin^2a-cos^2a=2-1=1\)

c/ \(C=sin^6x+cos^6x+3sin^2x.cos^2x\)

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

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

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

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

a) \(A=2\left(sin^6\alpha+cos^6\alpha\right)-3\left(sin^4\alpha+cos^4\alpha\right)\)
\(=2\left(sin^2\alpha+cos^2\alpha\right)\left(sin^4\alpha-sin^2\alpha cos^2\alpha+cos^4\alpha\right)\)\(-3\left(sin^4\alpha+cos^4\alpha\right)\)
\(=2\left(sin^4\alpha+cos^4\alpha-sin^2\alpha cos^2\alpha\right)-3\left(sin^4\alpha+cos^4\alpha\right)\)
\(=-\left(sin^4\alpha+cos^4\alpha+2sin^2\alpha cos^2\alpha\right)\)
\(=-\left(sin^2\alpha+cos^2\alpha\right)^2=-1\) (Không phụ thuộc vào \(\alpha\)).

b) \(B=4\left(sin^4\alpha+cos^4\alpha\right)-cos4\alpha\)
\(=4\left(sin^4\alpha+cos^4\alpha+2sin^2\alpha cos^2\alpha\right)-8sin^2\alpha cos^2\alpha\)\(-\left(1-2sin^22\alpha\right)\)
\(=4.\left(sin^2\alpha+cos^2\alpha\right)^2-2sin^22\alpha-1+2sin^22\alpha\)
\(=4-1=3\).

c) \(8\left(cos^8\alpha-sin^8\alpha\right)-cos6\alpha-7cos2\alpha\)
\(=8\left(cos^4\alpha-sin^4\alpha\right)\left(sin^4\alpha+cos^4\alpha\right)-cos6\alpha-7cos2\alpha\)
\(=8\left(cos^2\alpha-sin^2\alpha\right)\left(sin^4\alpha+cos^4\alpha\right)-cos6\alpha-7cos2\alpha\)
\(=8cos2\alpha\left(sin^4\alpha+cos^4\alpha\right)-cos6\alpha-7cos2\alpha\)
\(=8cos2\alpha\left(sin^4\alpha+cos^4\alpha\right)-8cos2\alpha+cos2\alpha-cos6\alpha\)
\(=8cos2\alpha\left(sin^4\alpha+cos^4-1\right)+sin4\alpha sin2\alpha\)
\(=8cos2\alpha\left[\left(sin^4\alpha-sin^2\alpha\right)+\left(cos^4\alpha-cos^2\alpha\right)\right]+\)\(sin4\alpha sin2\alpha\)
\(=8cos2\alpha.\left[sin^2\alpha\left(sin^2\alpha-1\right)+cos^2\alpha\left(cos^2\alpha-1\right)\right]\)\(+sin4\alpha sin2\alpha\)
\(=8.cos2\alpha.\left(-2sin^2\alpha cos^2\alpha\right)+2sin2\alpha cos2\alpha sin2\alpha\)
\(=-2cos2\alpha.\left(sin2\alpha\right)^2+2cos2\alpha.\left(sin2\alpha\right)^2\)
\(=sin^22\alpha\left(-cos2\alpha+cos2\alpha\right)=sin^22\alpha.0=0\) (không phụ thuộc vào \(\alpha\)).

a, Áp dụng PTG: \(BC=\sqrt{AB^2+AC^2}=25\)

Áp dụng HTL: \(BH=\dfrac{AB^2}{BC}=9\)

b, \(\sin\alpha+\cos\alpha=1,4\Leftrightarrow\left(\sin\alpha+\cos\alpha\right)^2=1,96\)

\(\Leftrightarrow\sin^2\alpha+\cos^2\alpha+2\sin\alpha\cdot\cos\alpha=1,96\\ \Leftrightarrow\sin\alpha\cdot\cos\alpha=\dfrac{1,96-1}{2}=\dfrac{0,96}{2}=0,48\)

\(\sin^4\alpha+\cos^4\alpha=\left(\sin^2\alpha+\cos^2\alpha\right)^2-2\sin^2\alpha\cdot\cos^2\alpha\\ =1^2+2\left(\sin\alpha\cdot\cos\alpha\right)^2=1+2\cdot\left(0,48\right)^2=1,4608\)