Sin4x.cos4x = 1/2

To solve the equation sin(4x)cos(4x) = 1/2, we can use the double angle formula for sine:

sin(2θ) = 2sin(θ)cos(θ)

Let's rewrite the equation using this formula:

2sin(4x)cos(4x) = 1/2

Now, let's divide both sides of the equation by 2:

sin(4x)cos(4x) = 1/4

Next, we can use the identity sin(2θ) = 2sin(θ)cos(θ) to rewrite the equation:

sin(8x) = 1/4

Now, we need to find the values of x that satisfy this equation. We can use the inverse sine function (sin^(-1)) to solve for x:

8x = sin^(-1)(1/4)

x = (1/8)sin^(-1)(1/4)

Therefore, the solution to the equation sin(4x)cos(4x) = 1/2 is x = (1/8)sin^(-1)(1/4).