Question

If (tanθ + sinθ) = m and (tanθ - sinθ) = n. prove that (m² - n²)² = 16 mn.

Answer 1

We have

(tan θ + sin θ) = m and (tan θ − sin θ) = n

Now, LHS = (m² − n²)² 

= [(tan θ + sin θ)² − (tan θ − sin θ)² ]² 

= [(tan² θ + sin² θ + 2 tan θ sin θ) − (tan² θ + sin² θ − 2 tan θ sin θ)]² 

= [(tan² θ + sin² θ + 2 tan θ sin θ − tan² θ − sin² θ + 2 tan θ sin θ)]² 

= (4 tan θ sin θ)² 

= 16 tan² θ sin² θ

= 16 mn

[(tan θ + sin θ)(tan θ − sin θ) = mn] 

∴ (m² − n²)(m² − n²)² = 16mn