$For 0\le x\le 2\pi ,$ Match the equations in List I to no.of solutions in List II
 

List I  (Trigonometric equation)	List II    (no.of solution)
$I) Ta{n}^{2}x+Co{t}^{2}x=2$	$a) 2$
$II) Si{n}^{2}x-Cosx=1/4$	$b) 0$
$III) 4Si{n}^{2}x+6Co{s}^{2}x=10$	$c)1$
$IV) Sinx=1$	$d) 4$

 

$$\begin{gathered} \mathrm{i}) {\mathrm{Tan}}^2\mathrm{x}+{\cot }^2\mathrm{x}=2 \mathrm{in}[0, 2\mathrm{\pi }] \\ {\mathrm{Tan}}^2\mathrm{x}=1 \Rightarrow \mathrm{x}=\mathrm{n\pi }\pm \frac{\mathrm{\pi }}{4} \\ \frac{\mathrm{\pi }}{4}, \frac{3\mathrm{\pi }}{4}, \frac{5\mathrm{\pi }}{4}, \frac{7\mathrm{\pi }}{4} \\ 4 \mathrm{solve} \\ \mathrm{ii}) {\sin }^2\mathrm{x}-\cos \mathrm{x}=\frac{1}{4} \\ 4-4{\cos }^2\mathrm{x}-\cos \mathrm{x}=1 \\ 4{\cos }^2\mathrm{x}-\cos \mathrm{x}-3=0 \\ (4\cos \mathrm{x}+3) (\cos \mathrm{x}-1)=0 \\ \cos \mathrm{x}=1 \\ \mathrm{x}=0, 2\mathrm{\pi } \\ 2 \mathrm{solve} \end{gathered}$$

$$\begin{gathered} \mathrm{iii}) 4{\sin }^2\mathrm{x}+6{\cos }^2\mathrm{x}=10 \\ \sin \mathrm{x}=1, \cos \mathrm{x}=1 \\ \mathrm{is} \mathrm{not} \mathrm{possible} \\ \text{'}0\text{'} \mathrm{solve} \\ \mathrm{iv}) \sin \mathrm{x}=1=\sin \frac{\mathrm{\pi }}{2} \\ \mathrm{x}=\frac{\mathrm{\pi }}{2} \end{gathered}$$