Question

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

Answer 1

(tan square + sin square+2tansin-tan square-sin square+2tansin) square
=4tansin square=16 tan square sin square
mn=(tan+sin)(tan-sin)
mn=tan square-sin square
=sin square/cos square-sin square
=(sin square-sin square cos square)/cos square
=sin square(1-cos square)/cos square (sin square+cos square=1)(sin square=1-cos square)
=(sin square * sin square)/cos square
=sin square/cos square*sin square
=tan square sin square
=So 16 tan square sin square
=16mn(already proved that mn =tan square sin square)
Hence proved

Answer 2

Tan + sin = m
Tan -sin = m
mn = (tan + sin) (tan -sin) = sin^4/cos^2
(m^2-n^2)^2 = 16sin^4/cos^2 = 16 mn
Hence proved