Question

(sin^4 theta + cos^4 theta / 1 - 2 sin^2 theta cos ^2 theta) = 1

Answer 1

1. YOUR ANSWER IS HERE ,...............

Answer 2

$\large\underline{\sf{Solution-}}$

Given Trigonometric function is

$\rm :\longmapsto\:\dfrac{ {sin}^{4} \theta + {cos}^{4} \theta}{1 - 2 {sin}^{2} \theta {cos}^{2} \theta}$

The numerator can be rewritten as

$\rm \:  =  \:\dfrac{ {( {sin}^{2} \theta) }^{2} + {( {cos}^{2} \theta)}^{2} }{1 - 2 {sin}^{2}\theta {cos}^{2}\theta}$

Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,

$\rm :\longmapsto\:\boxed{ \tt{ \: {x}^{2} + {y}^{2} = {(x + y)}^{2} - 2xy \: }}$

So, using this identity, we get

$\rm \:  =  \:\dfrac{ {[ {sin}^{2}\theta + {cos}^{2}\theta]}^{2} - 2 {sin}^{2}\theta {cos}^{2}\theta}{1 - 2 {sin}^{2}\theta {cos}^{2}\theta}$

Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,

$\boxed{ \tt{ \: {sin}^{2}x + {cos}^{2}x = 1 \: }}$

So, using this identity, we get

$\rm \:  =  \:\dfrac{ {[1]}^{2} - 2 {sin}^{2}\theta {cos}^{2}\theta}{1 - 2 {sin}^{2}\theta {cos}^{2}\theta}$

$\rm \:  =  \:\dfrac{ 1 - 2 {sin}^{2}\theta {cos}^{2}\theta}{1 - 2 {sin}^{2}\theta {cos}^{2}\theta}$

$\rm \:  =  \:1$

Hence,

$\red{\rm :\longmapsto\:\boxed{ \tt{ \: \dfrac{ {sin}^{4} \theta + {cos}^{4} \theta}{1 - 2 {sin}^{2} \theta {cos}^{2} \theta} = 1 \: }}}$

Additional Information:-

Relationship between sides and T ratios

sin θ = Opposite Side/Hypotenuse

cos θ = Adjacent Side/Hypotenuse

tan θ = Opposite Side/Adjacent Side

sec θ = Hypotenuse/Adjacent Side

cosec θ = Hypotenuse/Opposite Side

cot θ = Adjacent Side/Opposite Side

Reciprocal Identities

cosec θ = 1/sin θ

sec θ = 1/cos θ

cot θ = 1/tan θ

sin θ = 1/cosec θ

cos θ = 1/sec θ

tan θ = 1/cot θ

Co-function Identities

sin (90°−x) = cos x

cos (90°−x) = sin x

tan (90°−x) = cot x

cot (90°−x) = tan x

sec (90°−x) = cosec x

cosec (90°−x) = sec x

Fundamental Trigonometric Identities

sin²θ + cos²θ = 1

sec²θ - tan²θ = 1

cosec²θ - cot²θ = 1