Question

Cos2A/1+sin2A=tan(π/4-A)

Answer 1

Hello users....

We have to prove that

Cos2A/1+sin2A=tan(π/4-A)

Solution:-

As shown in attachment...

Formula used :

Cos 2x = cos²x- sin²x

Sin²x +cos² x =1

Sin 2x = 2sin x cos x

And

(1-tan x)/(1+tan x) = tan (π/4 -x)

⭐⭐ Hope it helps ⭐⭐

Answer 2

HELLO DEAR,

GIVEN:- cos2A/(1 + sin2A) = tan (π/4 - A)

taking L.H.S,

we know:-

cos2A = cos²A - sin²A

sin²A + cos²A = 1

tanx - tany/(1 + tanxtany) = tan(x - y)

now,

cos2A/(1 + sin2A)

=> (cos²A - sin²A)/(1 + 2sinAcosA)

=> (cos²A - sin²A)/(sin²A + cos²A + 2sinAcosA)

=> (cosA - sinA)(cosA + sinA)/(sinA + cosA)²

[ a² - b² = (a - b)(a + b) , (a + b)² = a² + b² + 2ab]

=> (cosA - sinA)/(cosA + sinA)

=> (1 - sinA/cosA)/(1 + sinA/cosA)

[dividing by cosA in num.,den.]

=> (1 - tanA)/(1 + tanA)

=> (tanπ/4 - tanA)/(1 + tanπ/4.tanA)

=> tan(π/4 - A)

HENCE, L H.S IS EQUAL TO R.H.S

I HOPE IT'S HELP YOU DEAR,

THANKS