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Question
Simplify `(1/("p") + 1/("q" + "r"))/(1/"p" - 1/("q" + "r")) xx [1 + ("q"^2 + "r"^2 - "p"^2)/(2"qr")]`
Solution
`1/"p" + 1/("q" + "r") = ("q" + "r" + "p")/("p"("q" + "r"))`
= `("p" + "q" + "r")/("p"("q" + "r"))`
`1/"p" - 1/("q" + "r") = ("q" + "r" - "p")/("p"("q" + "r"))`
`1 + ("q"^2 + "r"^2 - "p"^2)/(2"qr") = (2"qr" + "q"^2 + "r"^2 - "p"^2)/(2"qr")`
= `(("q" + "r")^2 - "p"^2)/(2"qr")`
= `(("q" + "r" + "p")("q" + "r" - "p"))/(2"qr")`
= `(("p" + "q" + "r")("q" + "r" - "p"))/(2"qr")`
`(1/("p") + 1/("q" + "r"))/(1/"p" - 1/("q" + "r")) xx [1 + ("q"^2 + "r"^2 - "p"^2)/(2"qr")]`
= `(("p" + "q" + "r"))/("p"("q" + "r")) xx ("p"("q" + "r"))/(("q" + "r" - "p")) xx (("p" + "q" + "r")("q" + "r" - "p"))/(2"qr")`
= `("p" + "q" + "r")^2/(2"qr")`
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