Question

证明(tan^2a+tana+1)(cot^2a+cota+1)=tan^2a+cot^2a+1

Answer 1

应该是证明:
(tan^2a+tana+1)(cot^2a+cota+1)=(tana+cota+1)²

证:
∵tana×cota=1
∴(tan²a+tana+1)(cot²a+cota+1)
=(tan²a+tana+1)×cot²a+(tan²a+tana+1)×cota+(tan²a+tana+1)×1
=1+cota+cot²a+tana+1+cota+tan²a+tana+1
=tan²a+cot²a+2+2cota+2tana+1
=(tan²a+cot²a+2×tanacota)+2cota+2tana+1
=(tana+cota)²+2(tana+cota)+1
=(tana+cota+1)²
得证.

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Answer 2

(tan^2a+tana+1)(cot^2a+cota+1)中应该是
(tan^2a+tana+1)(cot^2a-cota+1)=tan^2a+cot^2a+1
tan^2a+cot^2a+1
=tan^2a+1/tan^2a+1
=(tana+1/tana)^2-1
=(tana+1/tana+1)(tana+1/tana-1)
=(tan^2a+tana+1))[(tana+1/tana-1)/tana]
=(tan^2a+tana+1)(1+1/tan^2a-1/tana)
=(tan^2a+tana+1)(cot^2a-cota+1)