Which one of the following is a value of θ, if θ satisfies the equation tan 2θ tan 4θ - 1 = 0; \(0<\theta<\frac{\pi}{2} \) ?

Given:

tan 2θ tan 4θ - 1 = 0, \(0<\theta<\frac{\pi}{2} \)

Concept Used:

If tan A + tan B = 1

Then A + B = 90⁰

Calculation:

According to the question

⇒ tan 2θ tan 4θ - 1 = 0

⇒ tan 2θ tan 4θ = 1

⇒ 2θ + 4θ = 90⁰

⇒ θ = 15⁰ = \(\frac{\pi }{12}\)

∴ The value of θ is \(\frac{\pi }{12}\) which satisfies the equation tan 2θ tan 4θ - 1 = 0.

Alternate Method We have to check for which value of θ, \(0<θ<\frac{π}{2} \)

The equation tan 2θ tan 4θ - 1 = 0 satisfies

Option (1): θ = \(\frac{\pi }{12}\) = 15⁰

⇒ tan 2θ tan 4θ - 1 = tan 2(15⁰) tan 4(15⁰) - 1

⇒ tan 2θ tan 4θ - 1 = tan 30⁰ tan 60⁰ - 1

⇒ tan 2θ tan 4θ - 1 = (\(\frac{1}{√3}\)× √3) - 1 = 0

∴ The value of θ is \(\frac{\pi }{12}\) which satisfies the equation tan 2θ tan 4θ - 1 = 0.