1. The English translation is: "If \(x^2 = \sin^2 30° + 4\cot^2 45° - \sec^2 60°\), determine the value of \(x\)."

2. If \(x = \sin^2 30^\circ + 4 \cot^2 45^\circ - \sec^2 60^\circ\), find the value of \(x\).

3. If \((x + 1)\cot^2\frac{\pi}{2} = 2\cos^2\frac{\pi}{3} + \frac{3}{4}\sec^2\frac{\pi}{4} + 4\sin^2\frac{\pi}{6}\), then find the value of \(x\).

4. If \(x^2 = \sin^2 30^\circ + 4\cot^2 45^\circ - \sec^2 60^\circ\), find the value of \(x\). Let me know if you'd like a step-by-step solution next. I'm ready when you are.

5. If \(x^2 = \sin^2 30^\circ + 4\cot^2 45^\circ - \sec^2 60^\circ\), find the value of \(x\).

6. If \(x^2 = \sin^2 30^\circ + 4\cot^2 45^\circ - \sec^2 60^\circ\), then find the value of \(x\).

7. If \(x^2 = \sin^2 30^\circ + 4\cot^2 45^\circ - \sec^2 60^\circ\), then find the value of \(x\).

8. If \(x^2 = \sin^2 30^\circ + 4\cot^2 45^\circ - \sec^2 60^\circ\), then the value of \(x\) is —

(a) \(\pm 1\) (b) \(\pm \cfrac{1}{2}\) (c) \(\pm \cfrac{1}{\sqrt2}\) (d) \(\pm \cfrac{1}{\sqrt3}\)

9. If \( \sqrt{2} \sin(2x + 5^\circ) = \cot 45^\circ \), then what is the value of \( \sec 3x \)?

10. If \(\sin(2x + y) = \cos(4x - y)\), find the value of \(\tan 3x\).

11. Given: \(\tan A = \frac{x}{y}\), find the value of \(\frac{\cos A - \sin A}{\cos A + \sin A}\).

12. If \(x \sin 60^\circ \cos^2 30^\circ = \frac{\tan^2 45^\circ \sec 60^\circ}{\csc 60^\circ}\), find the value of \(x\).

13. If \( \csc \theta + \cot \theta = \sqrt{3} \), then find the value of \( \sin \theta \), where \( 0^\circ < \theta < 90^\circ \).

14. If \( \sin 23^\circ = p \), then express the value of \( \sin 67^\circ \) in terms of \( p \).

15. If \(\cot \theta = \cfrac{x}{y}\), then the value of \(\cfrac{x\cos\theta - y\sin\theta}{x\cos\theta + y\sin\theta}\) is —

(a) \(\cfrac{x^2+y^2}{x^2-y^2}\) (b) \(\cfrac{x^2}{x^2-y^2}\) (c) \(\cfrac{x^2}{x^2-y^2}\) (d) \(\cfrac{x^2-y^2}{x^2+y^2}\)

16. If \(x^2 = \sin^2 30^\circ + 4\cot^2 45^\circ - \sec^2 60^\circ\), then what is the value of \(x\)?

17. If \[ x \cot\left(\frac{\pi}{6}\right) = 2 \cos\left(\frac{\pi}{3}\right) + \frac{3}{4} \sec^2\left(\frac{\pi}{4}\right) + 4 \sin\left(\frac{\pi}{6}\right) \] then find the value of \(x\).

18. If \( x = \cfrac{\sqrt{a+2b}+\sqrt{a-2b}}{\sqrt{a+2b}+\sqrt{a-2b}} \), then determine the value of \( bx^2 - ax + b \).

(a) 1 (b) \(0\) (c) 3 (d) 4

19. \(y\) is directly proportional to the square root of \(x\), and \(y = 9\) when \(x = 9\). Determine the value of \(x\) when \(y = 6\).

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

20. 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) \(r^2\)

21. 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) \(\cfrac{225}{64}\)

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

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

23. If \( \csc A = \sqrt2 \), then find the value of
\(\cfrac{2\sin^2A + 3\cot^2A}{4\tan^2A - \cos^2A}\).

(a) \(\cfrac{8}{7}\) (b) \(\cfrac{7}{8}\) (c) \(\cfrac{1}{8}\) (d) \(\cfrac{1}{7}\)

24. In \( \triangle PQR \), ST is parallel to QR and intersects PQ at point S and PR at point T. Given that \( PQ = (X+1) \) units, \( PS = 3 \) units, \( PR = (X+6) \) units, and \( PT = 6 \) units, determine the value of \( x \).

25. If \(\alpha\) and \(\beta\) are the roots of the equation \(5x^2-3x+6=0\), determine the value of \(\left(\cfrac{1}{\alpha}+\cfrac{1}{\beta}\right)\).

26. If \(\alpha\) and \(\beta\) are the roots of the quadratic equation \(5x^2+2x+3=0\), determine the value of \(\cfrac{\alpha^2}{\beta}+\cfrac{\beta^2}{\alpha}\).

27. If \((5x - 3y) : (2x + 4y) = 11:12\), determine the value of \(x:y\).

28. In \(\triangle ABC\), the center of the incircle is \(O\), and the incircle touches the sides \(AB\), \(BC\), and \(CA\) at points \(P\), \(Q\), and \(R\) respectively. Given that \(AP = 4\) cm, \(BP = 6\) cm, \(AC = 12\) cm, and \(BC = x\) cm, determine the value of \(x\).

29. If the product of the roots of the equation \(x^2 - 3x + k = 10\) is \(-2\), determine the value of \(k\).

(a) -2 (b) -8 (c) 8 (d) 12

30. If one of the roots of the equations \(x^2 + bx + 12 = 0\) and \(x^2 + bx + q = 0\) is \(2\), determine the value of \(q\).