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If tan A = n tan B and sin A=msin B, pro...

If `tan A = n tan B` and `sin A=msin B,` prove that `cos^2 A =(m^2-1)/(n^2-1)`

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We have eliminate B. So, we find cosec B and cot B from the given relations and use the identity ` "cosec"^(2)B-cot^(2)B=1.`
`sinA= m sin B rArr "cosec"B=(m)/(sinA). " " `...(i)
`tan A=n tanB rArr cot B=(n)/(tanA). " " `...(ii)
Squaring (i) and (ii), and subtracting the results, we get
`(m^(2))/(sin^(2)A)-(n^(2))/(tan^(2)A)=1rArr m^(2)-n^(2)cos^(2)A=sin^(2)A rArr m^(2)-n^(2)cos^(2)A=1-cos^(2)A`
`rArr (n^(2)-1)cos^(2)A=(m^(2)-1) rArr cos^(2)A=((m^(2)-1))/((n^(2)-1)).`
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