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If 0ltalphaltpi/2 and sinalpha+cosalpha+...

If `0ltalphaltpi/2` and `sinalpha+cosalpha+tanalpha+cotalpha+secalpha+cosecalpha="7,` then prove that `sin2alpha` is a root of the equation `x^2-44x-36=0.`

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`sin alpha+(tan alpha+cot alpha)+(sec alpha+co sec alpha)=7`
or `(sin alpha+cos alpha)+(1)/(sin alpha cos alpha)+(sin alpha+cos alpha)/(sin alpha cos alpha)=7`
or `(sin alpha+cos alpha)(1+(1)/(sin alpha cos alpha))`
`=7-(1)/(sin alphacos alpha)`
Squaring both sides, we get
or `(1+sin 2alpha)(1+(4)/(sin 2 alpha)+(4)/(sin^(2)2alpha))`
`=49-(28)/(sin 2alpha)+(4)/(sin^(2)2alpha)`
Let `sin 2 alpha=x.` then
`(1+x)(1+(4)/(x)+(4)/(x^(2)))=49-(28)/(x)+(4)/(x^(2))`
or `(1+x)(x^(2)+4x+4)=49x^(2)-28x+4`
or `x^(3)-44x^(2)+36x=0`
or `x^(2)-44x+36=0 ("as" x=sin 2 alpha ne 0)`
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