Question

$\: \: \: \: \: \: \: \: \: \: \: \: \:$
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Answer 1

Step-by-step explanation:

please see the attachment

Answer 2

Identity Used :-

$(1). \: \boxed{ \blue{ \tt \: {x}^{2} + {y}^{2} = {(x + y)}^{2} - 2xy }}$

$(2). \: \boxed{ \blue{ \tt \: {sin}^{2}x + {cos}^{2}y = 1 }}$

$\large\underline\purple{\bold{Solution :- }}$

Consider,

$\bull \: \bf \: LHS$

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

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

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

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

$\rm :\implies\:1$

$\large{\boxed{\boxed{\bf{Hence, Proved}}}}$

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