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Question
If `m = (cos θ - sin θ)` and `n = (cos θ + sin θ)`, show that `sqrt(m/n) + sqrt(n/m) = 2/sqrt(1 - tan^2θ)`.
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Solution
LBS = `sqrt(m/n) + sqrt(n/m)`
= `sqrt(m)/sqrt(n) + sqrt(n)/sqrt(m)`
= `(m+n)/sqrt(mn)`
= `((cos theta - sin theta) + (cos theta + sin theta))/sqrt((cos theta - sin theta)(cos theta + sin theta))`
= `(2 cos theta)/ sqrt(cos ^2 theta - sin^2 theta)`
= `((( 2 cos theta)/(cos theta)))/((sqrt(cos^2 theta - sin^2 theta)/(cos theta))`
= `2/(sqrt((cos^2 theta)/(cos^2 theta) - (sin^2 theta)/(cos^2 theta))`
= `2/sqrt(1- tan^2 theta)`
= RHS
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