Question

If sinx + cosx = y, prove that sin^6x + cos^6x = [4-3(y^2-1)^2]/4

Answer 1

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

Given that

$\rm :\longmapsto\:sinx + cosx = y$

Identity used :-

$\red{\rm :\longmapsto\: {x}^{3} + {y}^{3} = {(x + y)}^{3} - 3xy(x + y)}$

$\red{\rm :\longmapsto\: {sin}^{2}x + {cos}^{2}x = 1}$

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

Consider LHS

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

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

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

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

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

$\bf\implies \: {sin}^{6}x + {cos}^{6}x=\: \: 1 - 3 {sin}^{2}x {cos}^{2}x - - (1)$

Now,

Consider RHS

$\rm :\longmapsto\:\dfrac{4 - 3 {( {y}^{2} - 1)}^{2} }{4}$

$\rm \: = \: \: \dfrac{4 - 3 {( {(sinx + cosx)}^{2} - 1)}^{2} }{4}$

$\rm \: = \: \: \dfrac{4 - 3 {( {sin}^{2}x + {cos}^{2}x + 2sinxcosx - 1)}^{2} }{4}$

$\rm \: = \: \: \dfrac{4 - 3 {( 1+2sinxcosx - 1)}^{2} }{4}$

$\rm \: = \: \: \dfrac{4 - 3 {(2sinxcosx )}^{2} }{4}$

$\rm \: = \: \: \dfrac{4 - 12 {sin}^{2}x {cos}^{2}x}{4}$

$\rm \: = \: \: \dfrac{4 (1- 3{sin}^{2}x {cos}^{2}x)}{4}$

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

$\bf :\longmapsto\:\dfrac{4 - 3 {( {y}^{2} - 1)}^{2} }{4} = 1 - {3sin}^{2}x {cos}^{2}x - - (2)$

From equation (1) and equation (2), we get

$\bf :\longmapsto\: {sin}^{6}x + {cos}^{6}x \: = \: \dfrac{4 - 3 {( {y}^{2} - 1)}^{2} }{4}$

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