Question

v) cos8 A - sin8 A = (1 - 2 sin2A cos2 A)(1-2 sin2A)​ please give a proper ans

Answer 1

Appropriate Question:-

Prove that

$\sf \: {cos}^{8}A - {sin}^{8}A = (1 - 2 {sin}^{2}A {cos}^{2}A)(1 - 2 {sin}^{2}A)$

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

Consider,

$\bf :\longmapsto\: {cos}^{8}A - {sin}^{8} A$

$\rm \: \: = \: \: {\bigg( {cos}^{4}A \bigg) }^{2} - {\bigg( {sin}^{4}A\bigg) }^{2}$

$\rm \: \: = \: \: ( {cos}^{4}A - {sin}^{4}A)( {cos}^{4}A + {sin}^{4}A)$

$\: \: \: \: \: \: \: \: \: \: \: \red{\bigg\{ \bf\because \: {x}^{2} - {y}^{2} = (x + y)(x - y)\bigg\}}$

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

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

$\: \red{\bigg\{ \bf\because \: {x}^{2} - {y}^{2} = (x + y)(x - y) \: and \: {x}^{2} + {y}^{2} = {(x + y)}^{2} - 2xy\bigg\}}$

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

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

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

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

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

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

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