1.
\(1+tan\alpha+tan^2\alpha+tan^3\alpha\)
\(=1+\dfrac{sin\alpha}{cos\alpha}+\dfrac{sin^2\alpha}{cos^2\alpha}+\dfrac{sin^3\alpha}{cos^3\alpha}\)
\(=1+\dfrac{sin\alpha}{cos\alpha}+\dfrac{sin^2\alpha}{cos^2\alpha}\left(1+\dfrac{sin\alpha}{cos\alpha}\right)\)
\(=\left(\dfrac{sin^2\alpha}{cos^2\alpha}+1\right)\left(1+\dfrac{sin\alpha}{cos\alpha}\right)\)
\(=\dfrac{1}{cos^2\alpha}\left(1+\dfrac{sin\alpha}{cos\alpha}\right)=\dfrac{sin\alpha+cos\alpha}{cos^3\alpha}\)
2.
\(\dfrac{1+tan^4x}{tan^2x+cot^2x}\)
\(=tan^2x.\dfrac{1+tan^4x}{tan^4x+cot^2x.tan^2x}\)
\(=tan^2x.\dfrac{1+tan^4x}{1+tan^4x}\)
\(=tan^2x\)
