7:
a: \(1+tan^2x=1+\left(\dfrac{sinx}{cosx}\right)^2\)
\(=\dfrac{cos^2x}{cos^2x}+\dfrac{sin^2x}{cos^2x}=\dfrac{1}{cos^2x}\)
b: \(1+cot^2x=1+\left(\dfrac{cosx}{sinx}\right)^2\)
\(=1+\dfrac{cos^2x}{sin^2x}=\dfrac{sin^2x+cos^2x}{sin^2x}=\dfrac{1}{sin^2x}\)
c: \(cos^4x-sin^4x\)
\(=\left(cos^2x-sin^2x\right)\left(cos^2x+sin^2x\right)\)
\(=cos^2x-sin^2x=2cos^2x-1\)
d: \(sin^6x+cos^6x\)
\(=\left(sin^2x+cos^2x\right)^3-3\cdot sin^2x\cdot cos^2x\left(sin^2x+cos^2x\right)\)
\(=1-3\cdot sin^2x\cdot cos^2x\)
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