4^(cos^2(x)) + 4^(cos^2(x)) = 3

Simplify:

2(4^(cos^2(x))) = 3

Divide by 2:

4^(cos^2(x)) = 1.5

Taking the natural logarithm of both sides:

ln(4^(cos^2(x))) = ln(1.5)

Using the property of logarithms:

cos^2(x) * ln(4) = ln(1.5)

Solving for cos^2(x):

cos^2(x) = ln(1.5) / ln(4)

cos^2(x) ≈ 0.7023

Now, take the square root of both sides to find cos(x):

cos(x) ≈ ±√0.7023

cos(x) ≈ ±0.8384

Therefore, the solutions for x are:

x ≈ arccos(0.8384) or x ≈ arccos(-0.8384)