Equations Solution

We are given the two trigonometric equations:

• Equation 1: \(p + q \cot \theta = 3 \cosec \theta\)
• Equation 2: \(q - p \cot \theta = 2 \cosec \theta\)

Trigonometric Transformation

To simplify, we can rewrite the equations using \(\sin \theta\) and \(\cos \theta\). Multiply both sides of each equation by \(\sin \theta\), assuming \(\sin \theta \neq 0\):

Equation 1 multiplied by \(\sin \theta\): \((p + q \frac{\cos \theta}{\sin \theta}) \sin \theta = 3 \frac{1}{\sin \theta} \sin \theta\) \(p \sin \theta + q \cos \theta = 3\) (Eq. 3)

Equation 2 multiplied by \(\sin \theta\): \((q - p \frac{\cos \theta}{\sin \theta}) \sin \theta = 2 \frac{1}{\sin \theta} \sin \theta\) \(q \sin \theta - p \cos \theta = 2\) (Eq. 4)

Squaring and Combining Equations

Now, square both Equation 3 and Equation 4:

Squaring Eq. 3: \((p \sin \theta + q \cos \theta)^2 = 3^2\) \(p^2 \sin^2 \theta + 2pq \sin \theta \cos \theta + q^2 \cos^2 \theta = 9\) (Eq. 5)

Squaring Eq. 4: \((q \sin \theta - p \cos \theta)^2 = 2^2\) \(q^2 \sin^2 \theta - 2pq \sin \theta \cos \theta + p^2 \cos^2 \theta = 4\) (Eq. 6)

Add the results from Eq. 5 and Eq. 6:

Combine like terms:

Identity \(p^2+q^2\) Result

Apply the fundamental trigonometric identity \(\sin^2 \theta + \cos^2 \theta = 1\):