Question

If tan θ + sin θ = m and tan θ − sin θ = n , the show that m2 − n2 = 4 √mn

Answer 1

Answer:

Step-by-step explanation:

m = Tanθ + Sinθ, n = Tanθ - Sinθ

L.H.S

m² - n² = (Tanθ + Sinθ)² - (Tanθ - Sinθ)² = 4TanθSinθ.

R.H.S

4√mn = 4√[(Tanθ + Sinθ)(Tanθ - Sinθ)]

           = 4√(Tan²θ - Sin²θ)

           = 4√[(Sin²θ/Cos²θ - Sin²θ)]

//Take Sin²θ common

           = 4√[Sin²θ ( 1/Cos²θ  - 1)]

           = 4√[Sin²θ(1 - Cos²θ/Cos²θ)]

           = 4√(Sin²θ * Sin²θ/Cos²θ )       ( ∵ 1 - Cos²θ = Sin²θ)

           = 4√(Sin⁴θ/Cos²θ)

           = 4Sin²θ/Cosθ

           = 4TanθSinθ

Hence L.H.S = R.H.S and thus proved.