Simplifying the Expression

To simplify the given expression, we need to apply trigonometric identities and properties. Let's break down the expression step by step:

Step 1: Simplify the expression inside the first tangent function.

The expression inside the first tangent function is:
3sin(2α) / (5 * 3cos(2α))

We can simplify this expression by canceling out the common factors of 3:
sin(2α) / (5cos(2α))

Step 2: Apply the quotient identity for tangent.

The quotient identity for tangent states that:
tan(A) = sin(A) / cos(A)

Using this identity, we can rewrite the expression as:
tan(2α) / 5

Step 3: Simplify the expression inside the second tangent function.

The expression inside the second tangent function is:
1 / (4tan(α))

We can simplify this expression by applying the reciprocal identity for tangent:
1 / (4 * sin(α) / cos(α))
cos(α) / (4sin(α))

Step 4: Combine the simplified expressions.

Now that we have simplified both parts of the expression, we can combine them:
(tan(2α) / 5) * (cos(α) / (4sin(α)))

Step 5: Simplify further if possible.

To simplify further, we can look for common factors to cancel out. In this case, we notice that both the numerator and denominator have a factor of sin(α):

(tan(2α) / 5) * (cos(α) / (4sin(α)))
(tan(2α) / 5) * (cos(α) / (4 * 1))
(tan(2α) / 5) * (cos(α) / 4)

Final Simplified Expression: tan(2α) * cos(α) / (20)

So, the simplified expression is tan(2α) * cos(α) / 20.