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Question
Prove the following identities.
`(sin "A" - sin "B")/(cos "A" + cos "B") + (cos "A" - cos "B")/(sin "A" + sin "B")`
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Solution
L.H.S = `(sin "A" - sin "B")/(cos "A" + cos "B") + (cos "A" - cos "B")/(sin "A" + sin "B")`
= `((sin "A" - sin "B")(sin "A" + sin "B") + (cos "A" - cos "B")(cos"A" + cos "B"))/((cos"A" + cos "B")(sin"A" + sin "B"))`
= `(sin^2"A" - sin^2"B" + cos^2"A" - cos^2"B")/((cos"A" + cos"B")(sin"A" + sin"B"))`
= `((sin^2"A" + cos^2"A") - (sin^2"B" + cos^2"B"))/((cos"A" + cos"B")(sin"A" + sin"B"))`
= `(1 - 1)/((cos"A" + cos"B")(sin"A" + sin"B")) = 0/((cos"A" + cos"B")(sin"A" + sin"B"))`
= 0
L.H.S = R.H.S ⇒ `(sin "A" - sin "B")/(cos "A" + cos "B") + (cos "A" - cos "B")/(sin "A" + sin "B")` = 0
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