Question:
if tan A - tan B = x and cot B -cot A =y, then prove that cot [A-B] = 1/x + 1/y .
Answer:
Given,
tan A - tan B = x ................1
and cot B - cot A = y ................2
From equation,
1/cot A - 1/cot B = x
=> (cot B - cot A)/(cot A * cot B) = x
=> y/(cot A * cot B) = x {from eqaution 2}
=> y = x * (cot A * cot B)
=> cot A * cot B = y/x
Now, cot (A - B) = (cot A * cot B + 1)/(cot B - cot A)
=> cot (A - B) = (y/x + 1)/y
=> cot (A - B) = (y/x)*(1/y) + 1/y
=> cot (A - B) = 1/x + 1/y
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