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Question
Prove that `(sec8 theta - 1)/(sec4 theta - 1) = (tan8 theta)/(tan2 theta)`
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Solution
We have `(sec8 theta - 1)/(sec4 theta - 1) = ((1 - cos8 theta) cos 4theta)/(cos 8theta(1 - cos4theta))`
= `(2sin^2 4theta cos 4theta)/(cos8theta 2sin^2 2theta)`
= `(sin4theta(2sin4theta cos 4theta))/(2cos8theta sin^2 2theta)`
= `(sin4theta sin 8theta)/(2 cos8 theta sin^2 2theta)`
= `(2sin 2theta cos 2theta sin 8theta)/(2 cos8theta sin^2 2theta)`
= `(tan8 theta)/(tan 2theta)`
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