Question

sin⁴A - cos⁴A = 2sin²A-1 prove that​

Answer 1

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

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