Question

2(sin⁶theta+cos⁶theta)-3(sin⁴theta+cos⁴theta)+1=0​

Answer 1

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

Formula Used :-

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

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

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

Consider, LHS

$\rm :\longmapsto\:2( {sin}^{6}\theta + {cos}^{6}\theta ) - 3( {sin}^{4}\theta + {cos}^{4}\theta ) + 1$

Consider,

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

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

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

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

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

Consider,

$\rm :\longmapsto\: {sin}^{4}\theta + {cos}^{4}\theta$

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

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

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

Now,

On substituting all these values in LHS, we get

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

$\rm \: = \: \:2 - 6{sin}^{2}\theta {cos}^{2} \theta ) -3 + 6{sin}^{2}\theta{cos}^{2} \theta + 1$

$\rm \: = \: \: 0$

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