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Prove that tan(pi/10) is a root of polyn...

Prove that `tan(pi/10)` is a root of polynomial equation `5x^4-10 x^2+1=0.`

Text Solution

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If `theta=18^(@)` then `5theta=90^(@)`
`rArr tan 5theta= oo`
`rArr (5 tan theta-.^(5)C_(3)tan ^(3)theta+.^(5)C_(5)tan^(5)theta)/(1-^(5)C_(2)tan ^(2)theta+^(5)C_(4)tan^(4)theta)=oo`
`rArr 5x^(4)-10x^(2)+1-0`, where `x=tan A`
Thus `x=tan theta` is root of the equation `5x^(4)-10x^(2)+1=0`
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