1. \((5 + \sqrt{3})(5 - \sqrt{3}) = 25 - x^2\); find the value of \(x\).

2. If \((\sqrt{5} + \sqrt{3})(\sqrt{5} - \sqrt{3}) = 25 - x^2\) is an equation, find the value of \(x\).

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

4. 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 \(x\).

5. If \[ x \cos 60^\circ = \frac{2 \tan 45^\circ}{1 + \tan^2 45^\circ} - \frac{1 - \tan^2 30^\circ}{1 + \tan^2 30^\circ} \] then find the value of \(x\).

6. 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 \(x\).

7. If \(x+\cfrac{1}{x}=\cfrac{13}{6}\), find the value of \(x\).

(a) \(\cfrac{2}{5}\) (b) \(1\cfrac{3}{4}\) (c) \(\cfrac{3}{2}\) (d) \(\cfrac{1}{4}\)

8. If \((\sqrt3 - \sqrt2)^x = (\sqrt3 + \sqrt2)^2\), then find the value of \(x\).

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

9. If \(\cfrac{x − x\tan^2 30°}{1 + \tan^2 30°} = \sin^2 30° + 4\cos^2 45° - \sec^2 60°\), find the value of \(x\). Let me know if you'd like it solved as well.

10. 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.

11. On the sides AC and BC of \(\triangle\)ABC, two points L and M are positioned respectively such that \(LM \parallel AB\), and \(AL = (x - 2)\) units, \(AC = 2x + 3\) units, \(BM = (x - 3)\) units, and \(BC = 2x\) units. Then, find the value of \(x\).

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

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

14. In the adjacent figure, if \(LM \parallel AB\), and \(AL = (x - 3)\) units, \(AC = 2x\) units, \(BM = (x - 2)\) units, and \(BC = (2x + 3)\) units, then find the value of \(x\).

15. If \[ 1\frac{1}{2} \cdot \frac{x + \sqrt{x^2 - 1}}{x - \sqrt{x^2 - 1}} + \frac{x - \sqrt{x^2 - 1}}{x + \sqrt{x^2 - 1}} = 14 \] then find the value of \(x\).

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

17. If \( (\sqrt{3} - \sqrt{2})^x = (\sqrt{3} + \sqrt{2})^2 \), then find the value of \(x\).

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