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Statement $-1:$ The number of common solutions of the trigonometric equations $2\,sin^2\,\theta - cos\,2\theta = 0$ and $2 \,cos^2\,\theta - 3\,sin\,\theta = 0$ in the interval $[0, 2\pi ]$ is two.
Statement $-2:$ The number of solutions of the equation, $2\,cos^2\,\theta - 3\,sin\,\theta = 0$ in the interval $[0, \pi ]$ is two.
Statement $-1$ is true; Statement $-2$ is true;Statement $-2$ is a correct explanation for statement $-1.$
Statement $-1$ is true; Statement $-2$ is true;Statement $-2$ is not a correct explanation for statement $-1.$
Statement $-1$ is false; Statement $-2$ is true.
Statement $-1$ is true; Statement $-2$ is false.
Solution
$2\,{\sin ^2}\theta \, – \,\cos \,2\theta \, = \,0$
$ \Rightarrow \,2\,{\sin ^2}\theta \, – \,(1 – 2\,{\sin ^2}\theta )\, = \,0$
$ \Rightarrow \,2\,{\sin ^2}\theta \, – \,1 + 2\,{\sin ^2}\theta \, = \,0$
$ \Rightarrow \,4\,{\sin ^2}\theta \, = 1 \Rightarrow \,\sin \theta \, = \, \pm \,\frac{1}{2}$
$\therefore \,\,\theta \, = \,\frac{\pi }{4},\frac{{3\pi }}{4},\frac{{5\pi }}{4},\frac{{7\pi }}{4},\,\theta \in \,\,[0\,,\,2\,\pi ]$
$\therefore \,\,\theta \, = \,\frac{\pi }{6},\frac{{5\pi }}{6},\frac{{7\pi }}{6},\frac{{11\pi }}{6}$
Now $2\,{\cos ^2}\,\theta \, – \,3\,\sin \,\theta = 0$
$ \Rightarrow \,\,2(1 – \,{\sin ^2}\,\theta )\, – \,3\,\sin \,\theta = 0$
$ \Rightarrow \, – \,2\,{\sin ^2}\,\theta \, – \,3\,\sin \,\theta \, + \,2 = 0$
$ \Rightarrow \, – \,2\,{\sin ^2}\,\theta \, – \,4\,\sin \,\theta + \sin \,\theta \, + \,2 = 0$
$ \Rightarrow \,2\,{\sin ^2}\,\theta \, – \,\,\sin \,\theta + 4\sin \,\theta \, – \,2 = 0$
$ \Rightarrow \sin \,\theta (\,2\,\sin \,\theta \, – 1) + 2(\,2\,\sin \,\theta \, – 1) = 0\,$
$ \Rightarrow \sin \,\theta \, = \frac{1}{2}\,,\, – \,2$
But $\sin \,\theta \, = \, – \,2\,$ , is not possible
$\therefore \,\,\sin \,\theta \, = \,\frac{1}{2},\,$ $ \Rightarrow \,\,\theta \, = \,\frac{\pi }{6},\frac{{5\pi }}{6}$
Hence, there are two common solution, there each of the statement $-1$ and $2$ are true but statement $-\,2$ is not a correct explanation for statment $-\,1$.
