To solve this question, we need to use the trigonometric identities for sine and cosecant.

Given: sin 450 = b cosec 300

To determine the value of a^4/b^4, we need to find the values of a and b.

Let's start by simplifying the given equation.

First, let's recall the values of sine and cosecant for the angles 450 and 300.

sin 450 = 1 (since the sine of 450 degrees is equal to the sine of 90 degrees, which is 1)

cosec 300 = 1/sin 300

Now, let's find the value of sin 300.

sin 300 = sin (360 - 60)
= sin 60
= √3/2

Therefore, cosec 300 = 1/(√3/2)
= 2/√3
= 2√3/3

Now, we can rewrite the given equation as:

1 = b * (2√3/3)

To isolate b, we divide both sides of the equation by (2√3/3):

1 / (2√3/3) = b

To simplify the left side, we multiply both the numerator and denominator by 3:

(1 * 3) / (2√3) = b

3 / (2√3) = b

Next, let's find the value of a.

Since we know that sin 450 = 1, we can write the equation as:

1 = a * (3 / (2√3))

To isolate a, we divide both sides of the equation by (3 / (2√3)):

1 / (3 / (2√3)) = a

To divide by a fraction, we multiply by its reciprocal:

1 * (2√3 / 3) = a

2√3 / 3 = a

Now that we have the values of a and b, we can find the value of a^4 / b^4.

a^4 / b^4 = (2√3 / 3)^4 / (3 / (2√3))^4

To simplify this expression, we can cancel out the common factors between the numerator and the denominator:

a^4 / b^4 = (2√3 / 3)^4 * ((2√3) / 3)^4

= (2√3 / 3)^4 * (2√3 / 3)^4

= (2√3 / 3)^8

= (2^8 * (√3)^8) / (3^8)

= (256 * 3^4) / (3^8)

= (256 * 81) / (3^8)

= 20736 / 6561

= 43

Therefore, the correct answer is option B: 43.