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if m=tanx+sinx and n=tanx-sinx then pr...

if `m=tanx+sinx and n=tanx-sinx` then prove that `m^2-n^2=4sqrt(mn)`

Text Solution

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Given: m =` tan x + sin x` and n= ` tan x - sin x`
L.H.S=` m^2 - n^2`
=` (tan x + sin x)^2 - ( tan x - sin x)^2`
= ` tan^2 x + 2 tan x sin x + sin^2 x - (tan^2 x - 2 tan x sin x + sin^2 x)`
= ` tan^2 x + 2 tan x sin x + sin^2 x - tan^2 x + 2 tan x sin x - sin^2 x`
= ` 4 tan x sin x`
R.H.S= ` 4 sqrt(mn)`
= ` 4 sqrt((tan x + sin x)(tan x - sin x))`
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