# Answer to Question #9819 in Trigonometry for lia

Question #9819

if tan a-tan b=p and cot b-cot a=q then cot(a-b)is?

Expert's answer

tan(a-b)={tan a-tan b}/{1+tan a tan b}

cot(a-b)=1/tan(a-b) => cot(a-b)={1+tan a tan b}/{tan a-tan b}={1+tan a tan b}/p

cot b-cot a=1/tan b -1/tan a={tan a - tan b}/{tan a tan b}=> tan a tan b={tan a - tan b}/{cot b-cot a}=p/q

So we get& cot(a-b)={1+p/q}/p

cot(a-b)=1/tan(a-b) => cot(a-b)={1+tan a tan b}/{tan a-tan b}={1+tan a tan b}/p

cot b-cot a=1/tan b -1/tan a={tan a - tan b}/{tan a tan b}=> tan a tan b={tan a - tan b}/{cot b-cot a}=p/q

So we get& cot(a-b)={1+p/q}/p

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