To simplify the left side of the equation, we need to find a trigonometric identity that can be applied to each term. We know that 1 + tan^2x = sec^2x.
So let's simplify each term using trigonometric identities:
sin^6x = (sin^2x)^3 = (1-cos^2x)^3 = 1 - 3cos^2x + 3cos^4x - cos^6x
cos^6x = (1-sin^2x)^3 = 1 - 3sin^2x + 3sin^4x - sin^6x
3cos^2x = 3(1-sin^2x) = 3 - 3sin^2x
Now substitute these simplifications back into the equation:
1 - 3cos^2x + 3cos^4x - cos^6x + 1 - 3sin^2x + 3sin^4x - sin^6x + 3 - 3cos^2x = 1 + tan^2x
Combine like terms and simplify:
6 - 6cos^2x + 3cos^4x + 3sin^4x - cos^6x - sin^6x = 1 + tan^2x
Now we cannot simplify any further, so the left side of the equation is:
6 - 6cos^2x + 3cos^4x + 3sin^4x - cos^6x - sin^6x
Therefore, the equation sin6x + cos6x + 3 - 3cos2x = 1 + tan2x cannot be simplified further.
To simplify the left side of the equation, we need to find a trigonometric identity that can be applied to each term. We know that 1 + tan^2x = sec^2x.
So let's simplify each term using trigonometric identities:
sin^6x = (sin^2x)^3 = (1-cos^2x)^3 = 1 - 3cos^2x + 3cos^4x - cos^6x
cos^6x = (1-sin^2x)^3 = 1 - 3sin^2x + 3sin^4x - sin^6x
3cos^2x = 3(1-sin^2x) = 3 - 3sin^2x
Now substitute these simplifications back into the equation:
1 - 3cos^2x + 3cos^4x - cos^6x + 1 - 3sin^2x + 3sin^4x - sin^6x + 3 - 3cos^2x = 1 + tan^2x
Combine like terms and simplify:
6 - 6cos^2x + 3cos^4x + 3sin^4x - cos^6x - sin^6x = 1 + tan^2x
Now we cannot simplify any further, so the left side of the equation is:
6 - 6cos^2x + 3cos^4x + 3sin^4x - cos^6x - sin^6x
Therefore, the equation sin6x + cos6x + 3 - 3cos2x = 1 + tan2x cannot be simplified further.