If tan θ + sin θ = m and tan θ – sin θ = n, then prove that

m² – n² = 4 sin θ tan θ

[Hint: m + n = 2tanθ, m – n = 2 sin θ, then use m² – n² = (m + n)(m – n)]

Given,

tan θ + sin θ = m …(1)

tan θ – sin θ = n …(2)

As we have to prove: m² – n² = 4 sin θ tan θ

∴ if we find out the expression of sin θ and tan θ in terms of m and n, we can get the desired expression to be proved.

∴ Adding equation 1 and 2 to get the value of tan θ.

2 tan θ = m + n …(3)

Similarly, on subtracting equation 2 from 1, we get-

2sin θ = m – n …(4)

Multiplying equation 3 and 4 –

2sin θ (2tan θ) = (m + n)(m – n)

⇒ 4 sin θ tan θ = m² – n²

Hence,

m² – n² = 4 sin θ tan θ