Question

show that: cos³A+sin³A/cosA+sinA + cos³A-sin³A/cosA-sinA=2​

Answer 1

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

Consider LHS

$\rm :\longmapsto\:\dfrac{ {cos}^{3}A + {sin}^{3}A}{cosA + sinA} + \dfrac{ {cos}^{3}A - {sin}^{3}A}{cosA - sinA}$

Consider,

$\red{\rm :\longmapsto\:\dfrac{ {cos}^{3}A + {sin}^{3}A}{cosA + sinA}}$

We know,

$\red{ \boxed{ \sf{ \: {x}^{3} + {y}^{3} = (x + y)( {x}^{2} + {y}^{2} - xy)}}}$

So, using this identity, we get

$\red{\rm \:  =  \: \dfrac{ \cancel{(cosA + sinA)} \: \: ( {cos}^{2}A + {sin}^{2}A - cosAsinA) }{ \cancel{cosA + sinA}} }$

$\red{\rm \:  =  \: {cos}^{2}A + {sin}^{2}A - sinAcosA}$

$\red{\rm \:  =  \: 1 - sinAcosA}$

Hence,

$\red{\bf :\longmapsto\:\dfrac{ {cos}^{3}A + {sin}^{3}A}{cosA + sinA} = 1 - sinAcosA}$

Now, Consider

$\purple{\rm :\longmapsto\:\dfrac{ {cos}^{3}A - {sin}^{3}A}{cosA - sinA}}$

We know,

$\purple{ \boxed{ \sf{ \: {x}^{3} - {y}^{3} = (x - y)( {x}^{2} + {y}^{2} + xy)}}}$

So, using this identity, we get

$\purple{\rm \:  =  \: \dfrac{ \cancel{(cosA - sinA)} \: \: ( {cos}^{2}A + {sin}^{2}A + cosAsinA) }{ \cancel{cosA - sinA}} }$

$\purple{\rm \:  =  \: {cos}^{2}A + {sin}^{2}A + sinAcosA}$

$\purple{\rm \:  =  \: 1 + sinAcosA}$

Hence,

$\purple{\bf :\longmapsto\:\dfrac{ {cos}^{3}A - {sin}^{3}A}{cosA - sinA} = 1 + sinAcosA}$

Now Consider,

$\rm :\longmapsto\:\dfrac{ {cos}^{3}A + {sin}^{3}A}{cosA + sinA} + \dfrac{ {cos}^{3}A - {sin}^{3}A}{cosA - sinA}$

On substituting the values evaluated above, we get

$\rm \:  =  \: 1 - sinAcosA + 1 + sinAcosA$

$\rm \:  =  \: 2$

Hence,

$\red{\bf\implies \: \boxed{ \sf{ \:\:\dfrac{ {cos}^{3}A + {sin}^{3}A}{cosA + sinA} + \dfrac{ {cos}^{3}A - {sin}^{3}A}{cosA - sinA} = 2}}}$

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