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If cosα/cosβ=m and cosα/sinβ=n show that (m^2+n^2)cos2β=n^2 - Mathematics

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Sum

`"If "\frac{\cos \alpha }{\cos \beta }=m\text{ and }\frac{\cos \alpha }{\sin \beta }=n " show that " (m^2 + n^2 ) cos^2 β = n^2`

 

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Solution

`LHS = (m^2 + n^2 ) cos^2 β`

`=( \cos ^{2}\alpha /\cos ^{2}\beta +\cos^{2}\alpha/\sin ^{2}\beta )\cos ^{2}\beta \text{}[ \because \ \ m=\frac{\cos \alpha }{\cos \beta} "and"n = cosalpha/sinbeta]`

`=(( \cos ^{2}\alpha \sin ^{2}\beta +\cos^{2}\alpha \cos ^{2}\beta)/( \cos ^{2}\beta \sin ^{2}beta))\cos ^{2}\beta`

`=\cos ^{2}\alpha ( \frac{1}{\cos ^{2}\beta \sin^{2}\beta ))\cos }^{2}\beta`

`=\cos ^{2}\alpha /\sin ^{2}\beta =( \frac{\cos\alpha }{\sin \beta })^{2}`

= n2 = RHS

Concept: Trigonometric Identities
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