Add `sin^4 x` both sides.

`cos^4 x=1+2sin^2 x+sin^4 x`

Use the formula: `a^2+2ab+b^2=(a+b)^2`

Compare `1+2sin^2 x+sin^4 x` and `a^2+2ab+b^2` and notice that a=1 and b=sin^2 x

Restrict `1+2sin^2 x+sin^4 x` using the formula.

`1+2sin^2 x+sin^4 x=(1+sin^2 x)^2`

Use the basic formula of trigonometry.

`sin^2 x+cos^2 x=1 =gtsin ^2 x = 1 - cos^2 x`

`(1+sin^2 x)^2 = (2-cos^2 x)^2`

If you expand the square `(2-cos^2 x)^2` you will obtain the result:

`(2-cos^2 x)^2 = 4 - 4cos^2 x + cos^4 x`

Notice that you did not get an identity.

**ANSWER: `cos^4 x - sin ^4 x!= 1+2sin^2 x.` **

**The expression is not identity.**