1. If \(0^\circ < \theta \leq 90^\circ\), then what is the minimum value of \((4\csc^2\theta + 9\sin^2\theta)\)?

2. If \(\tan \alpha = \cot \beta\), find the value of \(\cos(\alpha + \beta)\), where \(0^\circ < \alpha, \beta < 90^\circ\).

3. If \(2 \cos^2\theta + 3 \sin \theta = 3\) and \(0^\circ < \theta < 90^\circ\), find the value of \(\theta\).

4. If \(\sin A + \sin B = 2\), where \(0^\circ \leq A \leq 90^\circ\) and \(0^\circ \leq B \leq 90^\circ\), then find the value of \(\cos A + \cos B\).

5. If \(0^\circ < \theta < 90^\circ\), then determine the minimum value of \((9 \tan^2 \theta + 4 \cot^2 \theta)\).

6. Find the minimum value of: \[ 9\csc^2 \alpha + 25\sin^2 \alpha \quad \text{when} \quad 0^\circ \leq \alpha \leq 90^\circ \]

7. If \(0^\circ \le \theta \le 90^\circ\), then find the minimum value of \[ 9\tan^2\theta + 4\cot^2\theta \]

8. From the equation \(5 \sin^2 \theta + 4 \cos^2 \theta = \frac{9}{2}\), find the value of \(\tan \theta\), where \(0^\circ < \theta < 90^\circ\).

9. Find the minimum value of the expression \((4\csc^2 θ + 9\sin^2 θ)\) for \(0° < θ < 90°\).

10. If \(r \cos θ = 2\sqrt{3}\), \(r \sin θ = 2\), and \(0^\circ < θ < 90^\circ\), then find the values of \(r\) and \(θ\).

11. If \(0° \le \alpha < 90°\), then the minimum value of \(sec^2 \alpha + \cos^2 \alpha\) is 2.

12. If \( 0° < θ < 90° \), then find the minimum value of \( 9 \tan^2 θ + 4 \cot^2 θ \).

13. If \(\sin(A + B) = 1\) and \(\cos(A - B) = 1\), then find the value of \(\cot 2A\), given that \(0^\circ \leq (A + B) \leq 90^\circ\) and \(A \geq B\).

14. If \(\sec \theta = \csc \phi\), where \(0^\circ < \theta < 90^\circ\) and \(0^\circ < \phi < 90^\circ\), then the value of \(\sin(\theta + \phi)\) is 1.