Question

If tan x + sin x =m and tan x - sin x =n
show that
${m}^{2} - {n}^{2} = 4 \sqrt{mn}$

Answer 1

this the answer for that

Answer 2

L.H.S=m²-n²

=(tan x +sin x )² - (tan x - sin x)²

= tan²x + sin²x+2tanx.sinx -[tan²x + sin²x -2tanx.sinx]

=4tanx.sinx

R.H.S= 4√(mn)

=4√[(tanx+sinx)(tanx-sinx)]

=4√[tan²x-sin²x]

=4√[(sin²x/cos²x)-sin²x]

=4√[{sin²x-sin²x.cos²x}/cos²x]

=4√[{sin²x(1-cos²x)}/cos²x]

=4√[{sin²x.sin²x}/cos²x]

=4√[tan²x.sin²x]

=4tanx.sinx

therefore, L.H.S=R.H.S