(cos3x)^4+(sin3x)^4=cos(g(x))
find g(x)
To find g(x), we need to simplify the given equation and identify the corresponding trigonometric identity that matches the simplified equation.
Let's start by simplifying the equation:
(cos^3(x))^4 + (sin^3(x))^4 = cos(g(x))
Using the power rule of exponents, we can simplify the equation further:
cos^12(x) + sin^12(x) = cos(g(x))
Now, we have raised both cosine and sine to the fourth power, resulting in cos^12(x) and sin^12(x).
To identify a corresponding trigonometric identity, let's consider the Pythagorean identity:
sin^2(x) + cos^2(x) = 1
We can raise both sides of this equation to the sixth power while keeping in mind the relationship between cos^2(x) and sin^2(x):
(sin^2(x) + cos^2(x))^6 = 1
Expanding the equation using the binomial theorem:
1 + 6(sin^2(x))^1(cos^2(x))^5 + 15(sin^2(x))^2(cos^2(x))^4 + 20(sin^2(x))^3(cos^2(x))^3 + 15(sin^2(x))^4(cos^2(x))^2 + 6(sin^2(x))^5(cos^2(x))^1 + (sin^2(x))^6 = 1
Simplifying:
1 + 6sin^2(x)cos^10(x) + 15sin^4(x)cos^8(x) + 20sin^6(x)cos^6(x) + 15sin^8(x)cos^4(x) + 6sin^10(x)cos^2(x) + sin^12(x) = 1
Comparing this simplified equation with the original equation, we can see that they are equivalent:
sin^12(x) = sin^12(x)
Therefore, we can conclude that g(x) = 12x.
So, g(x) = 12x.