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Prove that tan18 is a root of the equation 5x410x2+1=0 and hence prove that tan2184 = 0.1056

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Solution

If θ=185θ=90 3θ=902θ
or tan3θ=cot2θor3tt313t2=1t22t
5t410t2+1=0 \, \, \, t2=1±0.8
ort2=1±.8944=1.8944=0.1056
t2=tan218 is +ive and less than 1.
tan45=1tan18<tan45ort<1 and +ive . The other value being > 1 is rejected

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