If cos^4 3 x - sin^4 3 x = cos (g(x)), then
g(x) = ??.
For part (b), use a^4-b^4=(a^2-b^2)(a^2+b^2)!
To solve for g(x), we need to simplify the left side of the equation cos^4(3x) - sin^4(3x) and express it in terms of cos(g(x)).
Using the identity a^4 - b^4 = (a^2 - b^2)(a^2 + b^2), we can rewrite the left side of the equation as:
cos^4(3x) - sin^4(3x) = (cos^2(3x) - sin^2(3x))(cos^2(3x) + sin^2(3x))
Since cos^2x + sin^2x = 1 (by the Pythagorean identity), we can simplify the expression further:
(cos^2(3x) - sin^2(3x))(cos^2(3x) + sin^2(3x)) = (cos^2(3x) - sin^2(3x))(1)
Next, we can use the identity cos(2x) = cos^2(x) - sin^2(x) to rewrite cos^2(3x) - sin^2(3x):
cos^2(3x) - sin^2(3x) = cos(2(3x))
Substituting this back into our expression, we have:
(cos^2(3x) - sin^2(3x))(1) = cos(2(3x))(1) = cos(6x)
Therefore, g(x) = 6x.