Question

prove the following
sin⁶A + cos⁶A = 1 - 3sin²A + 3sin⁴A​

Answer 1

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

Consider,

$\rm :\longmapsto\: {sin}^{6} A + {cos}^{6} A$

$\:  \: \rm  =  \:  \: {\bigg( {sin}^{2}A \bigg) }^{3} + {\bigg( {cos}^{2}A\bigg) }^{3}$

$\: \:\rm=\:\: {\bigg({sin}^{2}A + {cos}^{2}A\bigg) }^{3} -3{sin}^{2}A {cos}^{2}A({sin}^{2}A+{cos}^{2}A)$

$\: \: \: \: \: \red{ \boxed{ \because \bf \: {x}^{3} + {y}^{3} = {(x + y)}^{3} - 3xy(x + y)}}$

$\:  \: \rm  =  \:  \: {(1)}^{3} - 3 {sin}^{2}A {cos}^{2}A (1)$

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

$\:  \: \rm  =  \:  \: 1 - 3 {sin}^{2}A {cos}^{2}A$

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

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 2

Step-by-step explanation:

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

Consider,

$\rm :\longmapsto\: {sin}^{6} A + {cos}^{6} A$

$\:  \: \rm  =  \:  \: {\bigg( {sin}^{2}A \bigg) }^{3} + {\bigg( {cos}^{2}A\bigg) }^{3}$

$\: \:\rm=\:\: {\bigg({sin}^{2}A + {cos}^{2}A\bigg) }^{3} -3{sin}^{2}A {cos}^{2}A({sin}^{2}A+{cos}^{2}A)$

$\: \: \: \: \: \red{ \boxed{ \because \bf \: {x}^{3} + {y}^{3} = {(x + y)}^{3} - 3xy(x + y)}}$

$\:  \: \rm  =  \:  \: {(1)}^{3} - 3 {sin}^{2}A {cos}^{2}A (1)$

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

$\:  \: \rm  =  \:  \: 1 - 3 {sin}^{2}A {cos}^{2}A$

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

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