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Question
Show that : `(1)/(1 + "a"^("p"- "q")) + (1)/(1 + "a"^("q"- "p")`
Solution
L.H.S.
= `(1)/(1 + "a"^("p"- "q")) + (1)/(1 + "a"^("q"- "p")`
= `(1 + "a"^("q"- "p") + 1 + "a"^("p" - "q"))/((1 + "a"^("p"-"q"))(1 + "a"^("q"-"p"))`
= `(2 + "a"^-("p" -"q") + "a"^("p" - "q"))/((1 + "a"^("p"-"q"))(1 + "a"^-("q"-"p"))`
= `(2 + "a"^-("p"-"q") + "a"^("p"-"q"))/(1 + "a"^-("p"-"q") + "a"^("p"-"q") + "a"^("p"-"q") . "a"^-("p"-"q")`
= `(2 + "a"^-("p"-"q") + "a"^("p"-"q"))/(1 + "a"^-("p"-"q") + "a"^("p"-"q") + "a"^("p"-"q"-"p"+"q")`
= `(2 + "a"^-("p"-"q") + "a"^("p"-"q"))/(1 + "a"^-("p"-"q") + "a"^("p"-"q") + "a"^0`
= `(2 + "a"^-("p"-"q") + "a"^("p"-"q"))/(1 + "a"^-("p"-"q") + "a"^("p"-"q") + 1`
= `(2 + "a"^-("p"-"q") + "a"^("p"-"q"))/(2 + "a"^-("p"-"q") + "a"^("p"-"q"))`
= 1
= R.H.S.
Hence proved.
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