Question

if tan x + sin x = m and tan x - sin x = n then prove that m ^ 2 - n ^ 2 = 4sin x * tan x​

Answer 1

Step-by-step explanation:

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