a: \(tan^2a+cot^2a+2\)
\(=\dfrac{sin^2a}{cos^2a}+\dfrac{cos^2a}{sin^2a}+2\)
\(=\dfrac{sin^4a+cos^4a+2\cdot sin^2a\cdot cos^2a}{sin^2a\cdot cos^2a}=\dfrac{\left(sin^2a+cos^2a\right)^2}{sin^2a\cdot cos^2a}\)
\(=\dfrac{1}{sin^2a\cdot cos^2a}\)
b: \(tan^3a+tan^2a+tana+1\)
\(=\left(tana+1\right)\left(tan^2a+1\right)\)
\(=\left(\dfrac{sina}{cosa}+1\right)\cdot\left(\dfrac{sin^2a}{cos^2a}+1\right)\)
\(=\dfrac{sina+cosa}{cosa}\cdot\dfrac{sin^2a+cos^2a}{cos^2a}=\dfrac{sina+cosa}{cos^3a}\)
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