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

[Hint: Use the identities and ]

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,