We are given two trigonometric equations:

• \(\cosec \theta - \sin \theta = p^3\)
• \(\sec \theta - \cos \theta = q^3\)

We need to find the value of the expression \(p^4q^2 + p^2q^4\).

Simplifying Trigonometric Equations

First, let's simplify the equation for \(p^3\):

\(p^3 = \frac{1}{\sin \theta} - \sin \theta = \frac{1 - \sin^2 \theta}{\sin \theta}\)

Using the identity \(\sin^2 \theta + \cos^2 \theta = 1\), we get \(1 - \sin^2 \theta = \cos^2 \theta\).

Thus, \(p^3 = \frac{\cos^2 \theta}{\sin \theta}\).

Next, simplify the equation for \(q^3\):

\(q^3 = \frac{1}{\cos \theta} - \cos \theta = \frac{1 - \cos^2 \theta}{\cos \theta}\)

Using the identity \(\sin^2 \theta + \cos^2 \theta = 1\), we get \(1 - \cos^2 \theta = \sin^2 \theta\).

Thus, \(q^3 = \frac{\sin^2 \theta}{\cos \theta}\).

Calculating Powers of p and q

We need the terms \(p^4q^2\) and \(p^2q^4\). Let's find the necessary powers:

\(p^4 = (p^3)^{4/3} = \left( \frac{\cos^2 \theta}{\sin \theta} \right)^{4/3} = \frac{\cos^{8/3} \theta}{\sin^{4/3} \theta}\)

\(q^2 = (q^3)^{2/3} = \left( \frac{\sin^2 \theta}{\cos \theta} \right)^{2/3} = \frac{\sin^{4/3} \theta}{\cos^{2/3} \theta}\)

\(p^2 = (p^3)^{2/3} = \left( \frac{\cos^2 \theta}{\sin \theta} \right)^{2/3} = \frac{\cos^{4/3} \theta}{\sin^{2/3} \theta}\)

\(q^4 = (q^3)^{4/3} = \left( \frac{\sin^2 \theta}{\cos \theta} \right)^{4/3} = \frac{\sin^{8/3} \theta}{\cos^{4/3} \theta}\)

Evaluating the Target Expression

Now, calculate the two product terms:

\(p^4q^2 = \left( \frac{\cos^{8/3} \theta}{\sin^{4/3} \theta} \right) \left( \frac{\sin^{4/3} \theta}{\cos^{2/3} \theta} \right) = \cos^{(8/3 - 2/3)} \theta = \cos^2 \theta\)

\(p^2q^4 = \left( \frac{\cos^{4/3} \theta}{\sin^{2/3} \theta} \right) \left( \frac{\sin^{8/3} \theta}{\cos^{4/3} \theta} \right) = \sin^{(8/3 - 2/3)} \theta = \sin^2 \theta\)

Summing these terms:

\(p^4q^2 + p^2q^4 = \cos^2 \theta + \sin^2 \theta\)

Using the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\), the value is 1.