Question

If (tan theta+sin theta)=m and (tan theta- sin theta) =n prove that ( m^2-n^2)^2=16mn

Answer 1

 Tan θ + sinθ = m Squaring on both sides& Tan θ - Sin θ = n Squaring on both sides we get
prove that (m2 - n2)²=16mn

Tan2 θ + sin2 θ + 2 Tan θ sinθ = m2 ---------(1)

Tan2 θ + sin2 θ - 2 Tan θ sinθ = n2 --------(2)

Substract (2) from (1) we get

m2 - n2 = 4 Tan θ sinθ = 4 √( Tan2 θ  sin2 θ)
  = 4 √( Sin2 θ /Cos2 θ ( 1 - Cos2 θ) )
  = 4 √( Tan2 θ - sin2 θ)
= 4 √( Tan θ + Sin θ)(Tan θ - Sin θ )
m2 - n2 = 4√mn ( on squaring both sides)
(m2-n2)²=16mn

Answer 2

hope it will be helpful to you all