SOLUTION: cos^4A+Sin^2A=Sin^4A+Cos^2A

We start from equality

    cos^4A + Sin^2A = Sin^4A + Cos^2A.    (1)


We do not know, whether it is an identity.
It is our goal to PROVE that this equality is an identity, i.e. is valid for all values of angle A.
For it, we will perform equivalent transformations of this equality,
until we will get a valid identity.
It will allow us to conclude that the starting equality (1) is an identity.


Moving in (1) the term sin^4(A) from right side to left side (with changing its sign), 
and moving the term sin^2(A) from left side to right side (with changing its sign), 
we get an equivalent equality

    cos^4(A) - sin^4(A) = cos^2(A) - sin^2(A).    (2)


Equalities (1) and (2) are equivalent in that sense, that they EITHER both are true OR both are false.


Next, in (2) factor left side, using identity  a^2 - b^2 = (a+b)*(a-b) with a= cos^2(A), b= sin^2(B). 
You will get then this equality

    (cos^2(A) + sin^2(A))*(cos^2(A) - sin^2(A)) = cos^2(A) - sin^2(A).    (3)


Equalities (2) and (3) are equivalent in the same sense, that they EITHER both are true OR both are false.


But the expression in the first parentheses cos^2(A) + sin^2(A) is always 1, according to Pythagoras.
So, we replace this parentheses with 1, and we get from (3) 

    cos^2(A) - sin^2(A) = cos^2(A) - sin^2(A).    (4)


There is no need to repeat that equalities (3) and (4) are equivalent in the same sense as above.


        +-----------------------------------+
        |    But (4) is the TRUE identity.  |
        +-----------------------------------+


So, moving back from (4) to (3), we conclude that (3) is the true identity;

from it, moving back from (3) to (2), we conclude that (2) is the true identity; 

and from it, we conclude finally that (1) is the true identity.


In this way, the statement is proved.