Question

Verify sin^4x-sin^2x=cos^4x-cos^2x is an identity

Answer 1

ANSWER

See below

EXPLANATION

We want to verify that,

${ \sin ^{4} x} - { \sin^{2} x} = { \cos ^{4} x} - { \cos^{2} x}$

To verify this identity, we can take the left hand side simplify it to get the right hand side or vice versa.

${ \sin ^{4} x} - { \sin^{2} x} =( { \sin ^{2} x} )^{2} - { \sin^{2} x}$

${ \sin ^{4} x} - { \sin^{2} x} ={ \sin ^{2} x}({ \sin ^{2} x} - 1)$

${ \sin ^{4} x} - { \sin^{2} x} ={ \sin ^{2} x} \times - (1 - { \sin ^{2} x})$

${ \sin ^{4} x} - { \sin^{2} x} =({1 - \cos^{2} x} )\times - ({ \cos^{2} x})$

${ \sin ^{4} x} - { \sin^{2} x} =({ \cos^{2} x} - 1 )\times ({ \cos^{2} x})$

We now expand the right hand side to get:

${ \sin ^{4} x} - { \sin^{2} x} = { \cos ^{4} x} - { \cos^{2} x}$