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Question
Select the correct option from the given alternatives :
If tan A – tan B = x and cot B – cot A = y, then cot (A – B) = _____
Options
`1/y - 1/x`
`1/x - 1/y`
`1/x + 1/y`
`(xy)/(x - y)`
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Solution
If tan A – tan B = x and cot B – cot A = y, then cot (A – B) = `1/x + 1/y`
Explanation:
x = `sin"A"/cos"A" - sin"B"/cos"B"`
= `(sin"A"cos"B" - cos"A"sin"B")/(cos"A"cos"B")`
= `(sin("A" - "B"))/(cos"A" cos"B")`
y = `cos"B"/sin"B" - cos"A"/sin"A"`
= `(sin"A" cos"B" - cos"A"sin"B")/(sin"A"sin"B")`
= `(sin("A"- "B"))/(sin"A"sin"B")`
∴ `y/x = (cos"A"cos"B")/(sin"A"sin"B")` = cot A cot B
cot (A – B) = `(cot"A" cot"B" + 1)/(cot "B" - cot "A")`
= `(y/x + 1)/y`
= `(x + y)/(xy)`
= `1/x + 1/y`
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