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рдкреНрд░рд╢реНрди
Show that xcosecx = `1+x^2/6+(7x^4)/360+......`
рдЙрддреНрддрд░
LHS = xcosecx
=` x/sinx`
= `x/(x-x^3/(3!)+x^5/(5!)-x^6/(7!)+..........`
= `x/(x(1- x^2/(3!)+x^4/(5!)-x^6/(7!)+..........)`
=`[(1-(x^2/(3!)-x^4/(5!)+x^6/(7!)))]^(-1)`
=`1+(x^2/(3!)-x^4/(5!)+x^6/(7!)-..........)+(x^2/(3!)-x^4/(5!)+x^6/(7!)-.........)^2+.......{тИ╡(1-y)^1=1+y+y^2+y^3+.............}`
=`1+x^2/3!-x^4/5!+(x^2/3!)+........`
=`1+x^2/6+(7x^4)/360+........`
∴ xcosecx =` 1+x^2/6+(7x^4)/360+........`
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