Question

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

Answer 1

See below

Step-by-step 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} * - (1 - { \sin ^(2) x})`

`{ \sin ^(4) x} - { \sin^(2) x} =({1 - \cos^(2) x} )* - ({ \cos^(2) x})`

`{ \sin ^(4) x} - { \sin^(2) x} =({ \cos^(2) x} - 1 )* ({ \cos^(2) x})`

We now expand the right hand side to get:

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