`a) (1+sin^4\alpha-cos^4\alpha)/(1-sin^6\alpha-cos^6\alpha)`
`=(1+(sin^2\alpha-cos^2\alpha)(sin^2\alpha+cos^2\alpha))/(1-(sin^2\alpha+cos^2\alpha)(sin^4\alpha-sin^2\alpha.cos^2\alpha+cos^4\alpha))`
`=(1+sin^2\alpha-cos^2\alpha)/(1-[(sin^2\alpha+cos^2\alpha)^2-3sin^2\alpha.cos^2\alpha])`
`=(1+sin^2\alpha+cos^2\alpha-2cos^2\alpha)/(1-(1-3sin^2\alpha.cos^2\alpha))`
`=(2-2cos^2\alpha)/(1-1+3sin^2\alpha.cos^2\alpha)`
`=(2(1-cos^2\alpha))/(3sin^2\alpha.cos^2\alpha)`
`=(2.sin^2\alpha)/(3sin^2\alpha.cos^2\alpha)`
`=2/(3cos^2\alpha) (\text{đpcm})`
`b) (sin\alpha+tan\alpha)/(cos\alpha+cot\alpha)`
$=\dfrac{sin\alpha+\dfrac{sin\alpha}{cos\alpha}}{cos\alpha+\dfrac{cos\alpha}{sin\alpha}}$
$=\dfrac{\dfrac{sin^2\alpha.cos\alpha+sin^2\alpha}{sin\alpha.cos\alpha}}{\dfrac{cos^2\alpha.sin\alpha+cos^2\alpha}{sin\alpha.cos\alpha}}$
`=(sin^2\alpha(cos\alpha+1))/(cos^2\alpha(sin\alpha+1)) (\text{đpcm})`
`c) (tan\alpha-tan\beta)(cot\alpha-cot\beta)`
$=\dfrac{\dfrac{sin\alpha}{cos\alpha}-\dfrac{sin\beta}{cos\beta}}{\dfrac{cos\beta}{sin\beta}-\dfrac{cos\alpha}{sin\alpha}}$
$=\dfrac{\dfrac{sin\alpha.cos\beta-sin\beta.cos\alpha}{cos\alpha.cos\beta}}{\dfrac{cos\beta.sin\alpha-cos\alpha.sin\beta}{sin\alpha.sin\beta}}$
`=(sin\alpha.sin\beta)/(cos\alpha.cos\beta)`
`=(sin\alpha)/(cos\alpha).(sin\beta)/(cos\beta)`
`=tan\alpha.tan\beta (\text{đpcm})`
`d) (cos^2\alpha-sin^2\alpha)/(cot^2\alpha-tan^2\alpha)`
$=\dfrac{cos^2\alpha-sin^2\alpha}{\dfrac{cos^2\alpha}{sin^2\alpha}-\dfrac{sin^2\alpha}{cos^2\alpha}}$
$=\dfrac{cos^2\alpha-sin^2\alpha}{\dfrac{cos^4\alpha-sin^4\alpha}{sin^2\alpha.cos^2\alpha}}$
$=\dfrac{cos^2\alpha-sin^2\alpha}{\dfrac{(cos^2\alpha+sin^2\alpha)(cos^2\alpha-sin^2\alpha)}{sin^2\alpha.cos^2\alpha}}$
$=\dfrac{cos^2\alpha-sin^2\alpha}{\dfrac{cos^2\alpha-sin^2\alpha}{sin^2\alpha.cos^2\alpha}}$
`=sin^2\alpha.cos^2\alpha (\text{đpcm})`
