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If secx and tanx are the roots of the equation ax^2+bx +c =0 then prove that a^4-b^4+4ab^2c=0

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one year ago

Arun
25333 Points
```							ax^2 + bx + c=0  roots are secx & tanx  so product of roots - = secx*tanx = sinx /cos^2x = c/a--------------------------(I) sum of roots secx + tanx= - b/a---------------------(II) from (I)sinx/( 1- sin^2x) = c/a----------------------------(III) from(II) 1/cosx + sinx/cosx = -b/a (1+ sinx)^2/( cos^2x = b^2/a^2 1+ sinx/ 1- sinx = b^2/a2[componendo & dividendo ] or 1+ sinx -1+sinx/ ( 1+ sinx +1- sinx) = b^2- a^2/a^2+ b^2 or sinx= b^2- a^2/a^2+b^2-------------------(IV) from (III) 7 (IV) b^2- a^2/a^2+b^2/( 1 -{ (b^2-a^2)/(a^2+b^2)}^2 = c/a or (1 - (b^2-a^2/(b^2+a^2)}^2/ ( b^2-a^2/b^2+a^2)= a/c or [(b^2+a^2)^2 - ( b^-a^2)^2 /(b^2-a^2)( b^2+a^2) = a/c or[( b^2+a^2+b2-a^2)( b^2+a^2-b^2+ a^2)= a/c( b^4 - a^4) or 2 b^2 *2a^2 = a/c ( b^4 - a^4) or b^4 -a^4 - 4b^2ca =0 or b^4 = 4ab^2c + a^4 ANSWER
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one year ago
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