(cos3x)^4+(sin3x)^4=cos(g(x))

find g(x)

To find g(x), we need to simplify the given expression by using the trigonometric identity:

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

We can apply this identity to the given expression:

(cos^3(x))^4 + (sin^3(x))^4 = cos(g(x))

Let's simplify each term:

(cos^3(x))^4 = cos^12(x)

(sin^3(x))^4 = sin^12(x)

Now we can substitute these simplified terms into the equation:

cos^12(x) + sin^12(x) = cos(g(x))

Since cos^2(x) + sin^2(x) = 1, we can rewrite the equation as:

cos^12(x) + (1 - cos^2(x)) = cos(g(x))

Expanding the equation further:

cos^12(x) + 1 - cos^2(x) = cos(g(x))

Rearranging the terms:

cos^12(x) - cos^2(x) + 1 = cos(g(x))

Finally, comparing this equation to the original equation, we can conclude that g(x) = 12x.