Q. 233.6( 5 Votes )

If a cos2θ + b sin 2θ = c has α and β as its roots, then prove that

[Hint: Use the identities and ]

Answer :

Given,


a cos2θ + b sin 2θ = c and α and β are the roots of the equation.


Using the formula of multiple angles, we know that –


and



a(1 – tan2θ) + 2b tan θ - c(1 + tan2θ) = 0


(-c – a)tan2θ + 2b tan θ - c + a = 0 …(1)


Clearly it is a quadratic equation in tan θ and as α and β are its solutions.


tan α and tan β are the roots of this quadratic equation.


We know that sum of roots of a quadratic equation:


ax2 + bx + c = 0 is given by (-b/a)


tan α + tan β =


Hence,


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