tan(a)/[(1-cot(a)]+cot(a)/[1-(tan(a)]=1+tan(a)+cot(a)
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since cot(a)=1/tan(a)
tan(a)/[1-cot(a)]= -tan^a/[1-tan(a)]-----1
cot(a)/[1-tan(a)]= 1/[tan(a){1-tan(a)}]-----2
adding 1 and 2,
1-tan^3(a)/[tan(a){1-tan(a)}]
=(1-tan(a))(1+tan^2(a)+tan(a))/[tan(a){1-tan(a)}]
=(1+tan^2(a)+tan(a))/tan(a)
=1/tan(a) + tan^2(a)/tan(a) + tan(a)/tan(a)
=cot(a) + tan(a) + 1
hence proved
tan(a)/[1-cot(a)]= -tan^a/[1-tan(a)]-----1
cot(a)/[1-tan(a)]= 1/[tan(a){1-tan(a)}]-----2
adding 1 and 2,
1-tan^3(a)/[tan(a){1-tan(a)}]
=(1-tan(a))(1+tan^2(a)+tan(a))/[tan(a){1-tan(a)}]
=(1+tan^2(a)+tan(a))/tan(a)
=1/tan(a) + tan^2(a)/tan(a) + tan(a)/tan(a)
=cot(a) + tan(a) + 1
hence proved
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