Trigonometric equation with algebraic or logarithmic coupling

The question combines trigonometric equations with algebraic structures (quadratic equations with trig roots, logarithmic identities involving trig functions, or set-theoretic formulations) requiring joint reasoning.

jee-main 2013 Q67 View
Let $\mathrm { A } = \{ \theta : \sin ( \theta ) = \tan ( \theta ) \}$ and $\mathrm { B } = \{ \theta : \cos ( \theta ) = 1 \}$ be two sets. Then:
(1) $\mathrm { A } = \mathrm { B }$
(2) $A \not\subset B$
(3) $B \not\subset A$
(4) $A \subset B$ and $B - A \neq \phi$
jee-main 2025 Q3 View
Let $\mathrm{A} = \left\{ x \in (0, \pi) - \left\{ \frac{\pi}{2} \right\} : \log_{(2/\pi)} |\sin x| + \log_{(2/\pi)} |\cos x| = 2 \right\}$ and $\mathrm{B} = \{ x \geqslant 0 : \sqrt{x}(\sqrt{x} - 4) - 3|\sqrt{x} - 2| + 6 = 0 \}$. Then $\mathrm{n}(\mathrm{A} \cup \mathrm{B})$ is equal to:
(1) 4
(2) 8
(3) 6
(4) 2