a: \(sin^4x+cos^4x\)
\(=\left(sin^2x+cos^2x\right)^2-2\cdot sin^2x\cdot cos^2x\)
\(=1-2\cdot sin^2x\cdot cos^2x\)
b: \(sin^6x+cos^6x\)
\(=\left(sin^2x+cos^2x\right)^3-3\cdot sin^2x\cdot cos^2x\left(sin^2x+cos^2x\right)\)
\(=1-3\cdot sin^2x\cdot cos^2x\)
c: \(tan^2x-sin^2x=\dfrac{sin^2x}{cos^2x}-sin^2x\)
\(=sin^2x\left(\dfrac{1}{cos^2x}-1\right)=sin^2x\cdot\left(\dfrac{1-cos^2x}{cos^2x}\right)\)
\(=\dfrac{sin^2x}{cos^2x}\cdot sin^2x=tan^2x\cdot sin^2x\)
d: \(cot^2x-cos^2x=\dfrac{cos^2x}{sin^2x}-cos^2x\)
\(=cos^2x\left(\dfrac{1}{sin^2x}-1\right)=cos^2x\cdot\dfrac{1-sin^2x}{sin^2x}\)
\(=\dfrac{cos^2x}{sin^2x}\cdot cos^2x=cos^2x\cdot cot^2x\)
e: \(sin^4x-cos^4x=\left(sin^2x+cos^2x\right)\left(sin^2x-cos^2x\right)\)
\(=sin^2x-cos^2x\)
\(=sin^2x-\left(1-sin^2x\right)=2\cdot sin^2x-1\)
