Simplifying the given expression:

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

Step 1: Simplifying the first term:

The first term is tan^-1(3 sin 2 alpha / 5 3 cos 2 alpha). We can simplify it by using the trigonometric identity:
tan^-1(x) = sin^-1(x / √(1 + x^2))

Applying this identity, we have:
tan^-1(3 sin 2 alpha / 5 3 cos 2 alpha) = sin^-1((3 sin 2 alpha) / √(1 + (3 sin 2 alpha)^2 / (5 3 cos 2 alpha)^2))

Step 2: Simplifying the second term:

The second term is tan^-4(1/4 tan alpha). To simplify this, we need to use the property:
tan^-4(x) = (tan^2(x))^-2 = (1/cos^2(x))^-2 = cos^4(x)

Applying this property, we have:
tan^-4(1/4 tan alpha) = cos^4(1/4 tan alpha)

Step 3: Simplifying the overall expression:

Now, we can simplify the overall expression by substituting the simplified forms of the individual terms:

sin^-1((3 sin 2 alpha) / √(1 + (3 sin 2 alpha)^2 / (5 3 cos 2 alpha)^2)) * cos^4(1/4 tan alpha)

Explanation of the steps taken:

1. To simplify the expression, we used the trigonometric identities and properties.
2. In the first term, we used the identity tan^-1(x) = sin^-1(x / √(1 + x^2)).
3. In the second term, we used the property tan^-4(x) = cos^4(x).
4. Finally, we substituted the simplified forms of the individual terms into the overall expression.

Final simplified expression:

sin^-1((3 sin 2 alpha) / √(1 + (3 sin 2 alpha)^2 / (5 3 cos 2 alpha)^2)) * cos^4(1/4 tan alpha)