If (tan∅+sin∅)=m and (tan∅-sin∅)=n, prove that, (m^2-n^2)^2=16mn. Created by takshnaria58.

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

September 2023 1 9 Report

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

Answer:

Step-by-step explanation

m² = (tan∅+sin∅)²

= tan²∅ + sin² + 2sin∅tan∅

n² = (tan∅-sin∅)²

= tan²∅ + sin²∅ -2sin∅tan∅

m²- n² = 4sin∅tan∅ = 4sin²∅/cos∅

mn = (tan∅+sin∅)(tan∅+sin∅)

= tan²∅ - sin²∅

= sin²∅/cos²∅ - sin²∅

= (sin²∅ - sin²∅cos²∅)/cos²∅

= sin²∅(1-cos²∅)/(cos²∅)

= sin²∅×sin²∅/cos²∅

=sin⁴∅/cos²∅

(m²-n²)² = 16sin⁴∅/cos²∅ = 16 (sin⁴∅/cos²∅)

=16mn

proved