1. If \(sinθ−cosθ=0,\) \( (0°<θ<90°)\) and \(secθ+cosecθ=x\), then the value of \(x\) is—?

(a) \(1\) (b) \(2\) (c) \(\sqrt2\) (d) \(2\sqrt2\)

2. If \(r\cosθ = 2\sqrt{3}\), \(r\sinθ = 2\), and \(0° < θ < 90°\), then find the values of \(r\) and \(θ\).

3. If \( \cotθ = \cfrac{15}{8} \), then find the value of
\(\cfrac{(2+2\sinθ)(1-\sinθ)}{(1+\cosθ)(2-2\cosθ)}\).

(a) \(0\) (b) \(225\) (c) \(64\) (d) \(\cfrac{225}{64}\)

4. If \( \sinθ = \cfrac{a}{\sqrt{a^2+b^2}}; 0° < θ < 90° \), then find the value of \( \tanθ \).

(a) (\(\cfrac{b}{a}\) (b) \(b^2\) (c) \(\cfrac{a}{b}\) (d) \(\cfrac{a^2}{b^2}\)

5. If \( \sinθ – \cosθ = 0 \) (where \( 0° ≤ θ ≤ 90° \)) and \( \secθ + \cscθ = x \), then find the value of \( x \).

(a) 1 (b) 2 (c) \(\sqrt2\) (d) \(2\sqrt2\)

6. If sec 3θ = cosec 2θ and 0° < 5θ < 90°, then what is the value of θ?

7. If \(\frac{\sinθ + \cosθ}{\sinθ - \cosθ} = 7\), then what is the value of \(\tanθ\)?

8. If \(3\sinθ + 4\cosθ = 5\), then what is the value of \(4\sinθ - 3\cosθ\)?

9. If \( \sinθ - \cosθ = 0 \), then the value of \(θ\) is \(60°\).

10. If \( \sinθ + \cosθ = \sqrt2 \sin(90° – θ) \), then find the value of \( \cotθ \).

(a) \(\cfrac{\sqrt2}{3}\) (b) \(1\) (c) \(\sqrt2\) (d) \(\sqrt2+1\)

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

12. If \( \tan(θ + 15^\circ) = \sqrt{3} \), then find the value of \( \sinθ + \cosθ \).

13. If \( a \cosθ + b \sinθ = 4 \) and
\( a \sinθ - b \cosθ = 3 \), then \( a^2 + b^2 \) = ?

(a) 7 (b) 12 (c) 25 (d) 32

14. If \(\cfrac{\cosθ - \sinθ}{\cosθ + \sinθ} = \cfrac{\sqrt3 - 1}{\sqrt3 + 1}\), then find the value of \(θ\).

(a) \(60°\) (b) \(90°\ (c) \(120°\) (d) \(30°\)

15. If \(x = r\sinθ \cosϕ\), \(y = r\sinθ \sinϕ\), and \(z = r\cosθ\), then find the value of \(x^2 + y^2 + z^2\).

(a) \(r\) (b) \(5r\) (c) \(\sqrt{r}\) (d) \(r^2\)

16. If \( \tanθ = \cfrac{1}{\sqrt7} \), then find the value of
\(\cfrac{\csc^2θ - \sec^2θ}{\csc^2θ + \sec^2θ}\).

(a) \(\cfrac{3}{4}\) (b) \(\cfrac{1}{4}\) (c) \(\cfrac{2}{3}\) (d) \(\cfrac{2}{3}\)

17. If \( \tan 4θ \cdot \tan 6θ = 1 \), then determine the value of \( θ \) given that \( 0° < θ < 90° \).

(a) 5° (b) 4° (c) 9° (d) 3°

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

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

20. If \( \csc^2 θ = 2\cot θ \) and \( 0° < θ < 90° \), then find the value of \( θ \).

21. If \(\cfrac{\sinθ + \cosθ}{\sinθ - \cosθ} = \cfrac{3}{2}\), find the value of \(\cosθ\)

(a) \(\cfrac{1}{5}\) (b) \(\cfrac{3}{2}\) (c) \(\cfrac{1}{\sqrt{26}}\) (d) None of these

22. If \( \csc^2 θ = 2 \cot θ \), then find the value of \( θ \), where \( 0^\circ < θ < 90^\circ \).

23. If \(r\cosθ = 1\) and \(r\sinθ = \sqrt{3}\), then find the values of \(r\) and \(θ\).

24. If \( \tan θ + \cot θ = 2 \), then what is the value of \( \tan^7 θ + \cot^7 θ \)?

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

26. If \(2\cos^2θ + 3\sinθ - 3 = 0\), then find the value of \(θ\).

(a) \(90°\) (b) \(30°\) (c) both (a) and (b) (d) none of the above

27. If \( \cosθ = \cfrac{3}{5} \), then find the value of \( \cotθ + \cscθ \).

(a) 1 (b) 2 (c) 3 (d) 4

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

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

30. 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.