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Question
Prove the following identities.
`(sin^3"A" + cos^3"A")/(sin"A" + cos"A") + (sin^3"A" - cos^3"A")/(sin"A" - cos"A")` = 2
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Solution
`(sin^3"A" + cos^3"A")/(sin"A" + cos"A") + (sin^3"A" - cos^3"A")/(sin"A" - cos"A")` = 2
L.H.S = `(sin^3"A" + cos^3"A")/(sin"A" + cos"A") + (sin^3"A" - cos^3"A")/(sin"A" - cos"A")`
= `((sin"A" + cos"A")(sin^2"A" - sin"A"cos"A" + cos^2"A"))/((sin"A" + cos"A")) + ((sin"A" - cos"A")(sin^2"A" + sin"A"cos"A" + cos^2"A"))/((sin"A" - cos"A"))`
= (sin2 A + cos2 A) − sin A cos A + (sin2 A + cos2 A) + sin A cos A
= 1 + 1
= 2
L.H.S = R.H.S
∴`(sin^3"A" + cos^3"A")/(sin"A" + cos"A") + (sin^3"A" - cos^3"A")/(sin"A" - cos"A")` = 2
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