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 \(x = r\sinθ \cosϕ\), \(y = r\sinθ \sinϕ\), and \(z = r\cosθ\), then find the value of \(x^2 + y^2 + z^2\).
(a) \(r\) (b) \(5r\) (c) \(\sqrt{r}\) (d) \(r^2\)
3. If sinθ + cosθ = √2 (where 0° < θ < 90°), then the value of θ is
(a) 30° (b) 45° (c) 60° (d) 90°
4. If sec 3θ = cosec 2θ and 0° < 5θ < 90°, then what is the value of θ?
5. If \(r\cosθ = 2\sqrt{3}\), \(r\sinθ = 2\), and \(0° < θ < 90°\), then find the values of \(r\) and \(θ\).
6. If \(r\cosθ = 1\) and \(r\sinθ = \sqrt{3}\), then find the values of \(r\) and \(θ\).
7. If \(\frac{\sinθ + \cosθ}{\sinθ - \cosθ} = 7\), then what is the value of \(\tanθ\)?
8. If \( \cos^2 θ - \sin^2 θ = \frac{1}{2} \), then find the value of \( \cos^4 θ - \sin^4 θ \).
9. If \( \cos^2θ − \sin^2θ = \frac{1}{x} \), where \( x > 1 \), then what is the value of \( \cos^4θ − \sin^4θ \)?
10. If \( \tan(θ + 15^\circ) = \sqrt{3} \), then find the value of \( \sinθ + \cosθ \).
11. If \(tan 35° \cdot tan 55° = \sin θ\), then the smallest positive value of \(θ\) will be ——.
12. If \( \cos^2 θ - \sin^2 θ = \cfrac{1}{2} \), then find the value of \( \tan^2 θ \).
13. If \(3\sinθ + 4\cosθ = 5\), then what is the value of \(4\sinθ - 3\cosθ\)?
14. If \( \sinθ - \cosθ = 0 \), then the value of \(θ\) is \(60°\).
15. 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
16. If \(\cfrac{\cosθ - \sinθ}{\cosθ + \sinθ} = \cfrac{\sqrt3 - 1}{\sqrt3 + 1}\), then find the value of \(θ\).
(a) \(60°\) (b) \(90°\ (c) \(120°\) (d) \(30°\)
17. If \( \cosθ = \cfrac{3}{5} \), then find the value of \( \cotθ + \cscθ \).
(a) 1 (b) 2 (c) 3 (d) 4
18. 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) \(\cfrac{225}{64}\)
19. If \( \sinθ + \cosθ = \sqrt2 \sin(90° – θ) \), then find the value of \( \cotθ \).
(a) \(\cfrac{\sqrt2}{3}\) (b) \(1\) (c) \(\sqrt2\) (d) \(\sqrt2+1\)
20. If \( \sinθ = \cos2θ \), then find the value of \( θ \).
(a) \(30°\) (b) \(45°\) (c) \(60°\) (d) \(90°\)
21. If \( r\cosθ = 2\sqrt{3} \), \( r\sinθ = 2 \), and \( 0^\circ < θ < 90^\circ \), then find the values of \( r \) and \( θ \).
22. If \(r \cos θ = 2\sqrt{3}\), \(r \sin θ = 2\), and \(0^\circ < θ < 90^\circ\), then find the values of \(r\) and \(θ\).
23. If \(\tan θ = 1\), then find the value of \(\frac{8 \sin θ + 5 \cos θ}{\sin^3 θ - 2 \cos^3 θ + 7 \cos θ}\).
24. If \(\sin θ + \cos θ = 1\), then find the value of \(\sin θ × \cos θ\).
25. If \(\sin θ - \cos θ = \frac{7}{13}\), then find the value of \(\sin θ + \cos θ\).
26. If \(\sin θ \cos θ = \frac{1}{2}\), then calculate the value of \((\sin θ + \cos θ)\).
27. If \(\frac{\sin θ + \cos θ}{\sin θ - \cos θ} = 7\), then calculate the value of \(\tan θ\).
28. If \(\sec θ + \cos θ = \frac{5}{2}\), then calculate the value of \(\sec θ - \cos θ\).
29. If \(5\sin^2 θ + 4\cos^2 θ = \frac{9}{2}\), then find the value of \(\tan θ\).
30. If \(\tan^2 θ + \cot^2 θ = \frac{10}{3}\), then find the values of \(\tan θ + \cot θ\) and \(\tan θ - \cot θ\), and from there calculate the value of \(\tan θ\).
