Question

If tan theta and sec theta the roots of the equation x square + bx + c equal to zero then

Answer 1

Answer:

Step-by-step explanation:

Since tanΘ and secΘ are the roots of the equation $ax^{2} +bx+c = 0$

Then,

tanΘ + secΘ= -b/a  -----------------(1)

tan Θ. secΘ= c/a  ---------------(2)

From (1) we get,

sinΘ/cosΘ + 1/cosΘ = -b/a

sinΘ+1/cosΘ = -b/a

Squaring...

(1+ sinΘ)^2/( cos^2Θ = b^2/a^2  

1+ sinΘ/ 1- sinΘ = b^2/a^2

or 1+ sinΘ -1+sinΘ/ ( 1+ sinΘ +1- sinΘ) = b^2- a^2/a^2+ b^2  

or sinΘ= 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