Sin⁴A - cos⁴A = 2sin²A-1 prove that

1.	Sin⁴A - cos⁴A = 2sin²A-1 prove that
Answer» $\large\underline{\sf{Solution-}}$ Identities used :- $\boxed{ \bf{ \: {x}^{2} - {y}^{2} = (x + y)(x - y)}}$ $\boxed{ \bf{ \: {sin}^{2} x + {cos}^{2} x = 1}}$ LET's solve the problem now!! Consider, $\rm :\longmapsto\: {sin}^{4}A - {cos}^{4}A$ $\rm :\longmapsto\: = \: {\bigg( {sin}^{2}A \bigg) }^{2} - {\bigg( {cos}^{2}A \bigg) }^{2}$ $\rm :\longmapsto\: = \: ( {sin}^{2} A + {cos}^{2} A)( {sin}^{2} A - {cos}^{2} A)$ $\rm :\longmapsto\: = \: 1 \times \bigg( {sin}^{2} A - (1 - {sin}^{2}A) \bigg)$ $\rm :\longmapsto\: = \: 2 {sin}^{2} A - 1$ ${\boxed{\boxed{\bf{Hence, Proved}}}}$ 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

Answer»

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

Identities used :-

$\boxed{ \bf{ \: {x}^{2} - {y}^{2} = (x + y)(x - y)}}$

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

LET's solve the problem now!!

Consider,

$\rm :\longmapsto\: {sin}^{4}A - {cos}^{4}A$

$\rm :\longmapsto\: = \: {\bigg( {sin}^{2}A \bigg) }^{2} - {\bigg( {cos}^{2}A \bigg) }^{2}$

$\rm :\longmapsto\: = \: ( {sin}^{2} A + {cos}^{2} A)( {sin}^{2} A - {cos}^{2} A)$

$\rm :\longmapsto\: = \: 1 \times \bigg( {sin}^{2} A - (1 - {sin}^{2}A) \bigg)$

$\rm :\longmapsto\: = \: 2 {sin}^{2} A - 1$

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

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