1. 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 \(x\).

2. 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) \(\cfrac{2}{3}\)

3. 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) \(\cfrac{1}{7}\)

4. If \(\sec θ + \tan θ = 2\), then find the value of \(\sec θ - \tan θ\).

5. If \(\tan θ + \cot θ = 2\), then find the value of \(\tan θ - \cot θ\).

6. 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 \(\tan θ\).

7. If \( \tan α + \cot α = 2 \), then find the value of \( \tan^{13} α + \cot^{13} α \).

(a) 1 (b) 0 (c) 2 (d) None of the above

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

9. If \(\cot θ = 2\), then find the values of \(\tan θ\) and \(\sec θ\), and show that: \[1 + \tan^2θ = \sec^2θ\]

10. If \( \tan\theta - \cot\theta = 0 \), then find the value of \( \sin(\theta - 15^\circ) + \cos(\theta + 15^\circ) \).

11. If \( \tan^2\theta + \cot^2\theta = \cfrac{10}{3} \), then find the values of \( \tan\theta + \cot\theta \) and \( \tan\theta - \cot\theta \). From there, determine the value of \( \tan\theta \).

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

13. If \(∠A + ∠B = 90°\), then find the value of \(1 + \frac{\tan A}{\tan B}\).

14. If \(∠A + ∠B = 90^\circ\), then find the value of \(1 + \tan A \div \tan B\).

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

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

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

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

19. If \( \frac{\sin\theta + \cos\theta}{\sin\theta - \cos\theta} = 5 \), then find the value of \( \tan\theta \).

20. If \[ \frac{\sec\theta + \tan\theta}{\sec\theta - \tan\theta} = 2\frac{51}{79} \] then find the value of \(\sin\theta\).

21. 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 \(x\).

22. If \(\sec \theta - \tan \theta = \frac{1}{2}\), then find the values of \(\sec \theta\) and \(\tan \theta\).

23. If \( \cos\theta = \frac{p}{\sqrt{p^2 + q^2}} \), then find the value of \( \tan\theta \).

(a) \(\cfrac{q}{p}\) (b) \(\cfrac{p}{q}\) (c) \(\cfrac{\sqrt{p}}{q}\) (d) None of the above

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

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

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

26. 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) \(\cfrac{a^2}{b^2}\)

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

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

28. If \( \sin \theta - \cos \theta = 0 \) (where \( 0^\circ \lt \theta \lt 90^\circ \)) and \( \sec \theta + \csc \theta = x \), then find the value of \( x \).

(a) 1 (b) 2 (c) √2 (d) 2√2

29. 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.

30. 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.