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\(sinx+cosx=m\Leftrightarrow\left(sinx+cosx\right)^2=m^2\)
\(\Leftrightarrow1+2sinx.cosx=m^2\Rightarrow sinx.cosx=\dfrac{m^2-1}{2}\)
\(A=sin^2x+cos^2x=1\)
\(B=sin^3x+cos^3x=\left(sinx+cosx\right)^3-3sinx.cosx\left(sinx+cosx\right)\)
\(=m^3-\dfrac{3m\left(m^2-1\right)}{2}=\dfrac{2m^3-3m^3+3m}{2}=\dfrac{3m-m^3}{2}\)
\(C=\left(sin^2+cos^2x\right)^2-2\left(sinx.cosx\right)^2=1-2\left(\dfrac{m^2-1}{2}\right)^2\)
\(D=\left(sin^2x\right)^3+\left(cos^2x\right)^3=\left(sin^2x+cos^2x\right)^3-3\left(sin^2x+cos^2x\right)\left(sinx.cosx\right)^2\)
\(=1-3\left(\dfrac{m^2-1}{2}\right)^2\)
\(\frac{cos^3x-cos3x}{cosx}+\frac{sin^3x+sin3x}{sinx}=cos^2x-\frac{cos3x}{cosx}+sin^2x+\frac{sin3x}{sinx}\)
\(=1+\frac{sin3x.cosx-cos3x.sinx}{sinx.cosx}=1+\frac{sin\left(3x-x\right)}{\frac{1}{2}sin2x}=1+\frac{2sin2x}{sin2x}=3\)
\(cot^2x-cos^2x=\frac{cos^2x}{sin^2x}-cos^2x=cos^2x\left(\frac{1}{sin^2x}-1\right)=\frac{cos^2x\left(1-sin^2x\right)}{sin^2x}\)
\(=cos^2x.\left(\frac{cos^2x}{sin^2x}\right)=cot^2x.cos^2x\)
\(\frac{cosx+sinx}{cosx-sinx}-\frac{cosx-sinx}{cosx+sinx}=\frac{\left(cosx+sinx\right)^2-\left(cosx-sinx\right)^2}{\left(cosx-sinx\right)\left(cosx+sinx\right)}\)
\(=\frac{cos^2x+sin^2x+2sinx.cosx-\left(cos^2x+sin^2x-2sinx.cosx\right)}{cos^2x-sin^2x}=\frac{4sinx.cosx}{cos2x}=\frac{2sin2x}{cos2x}=2tan2x\)
\(\frac{sin4x+cos2x}{1-cos4x+sin2x}=\frac{2sin2x.cos2x+cos2x}{1-\left(1-2sin^22x\right)+sin2x}=\frac{cos2x\left(2sin2x+1\right)}{sin2x\left(2sin2x+1\right)}=\frac{cos2x}{sin2x}=cot2x\)
\(A=sin^2x\left(sinx+cosx\right)+cos^2x\left(sinx+cosx\right)\)
\(=\left(sin^2x+cos^2x\right)\left(sinx+cosx\right)=sinx+cosx\)
\(B=\frac{sinx}{cosx}\left(\frac{1+cos^2x-sin^2x}{sinx}\right)=\frac{sinx}{cosx}\left(\frac{2cos^2x}{sinx}\right)=2cosx\)
\(\frac{cos^3x-cos3x}{cosx}+\frac{sin^3x+sin3x}{sinx}=cos^2x-\frac{cos3x}{cosx}+sin^2x+\frac{sin3x}{sinx}\)
\(=1+\frac{sin3x}{sinx}-\frac{cos3x}{cosx}=1+\frac{sin3x.cosx-cos3x.sinx}{sinx.cosx}\)
\(=1+\frac{sin\left(3x-x\right)}{\frac{1}{2}sin2x}=1+\frac{2sin2x}{sin2x}=1+2=3\)
sin x+cosx=m
=>(sinx+cosx)^2=m^2
=>1+2*cosx*sinx=m^2
=>2*sinx*cosx=m^2-1
=>\(sinx\cdot cosx=\dfrac{m^2-1}{2}\)
\(sin^3x+cos^3x=\left(sinx+cosx\right)^3-3\cdot sinx\cdot cosx\cdot\left(sinx+cosx\right)\)\(=m^3-3\cdot\dfrac{m^2-1}{2}\cdot m\)
\(=m^3-\dfrac{3m^3-3m}{2}\)
\(=\dfrac{2m^3-3m^3+3m}{2}=\dfrac{-m^3+3m}{2}\)