The common roots of the equations 2sin^2 x + | vedclass

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(B) Given equations are:$2\sin^2 x + \sin^2 2x = 2$ ... $(i)$$\sin 2x + \cos 2x = 	an x$ ... $(ii)$From $(i)$:$2\sin^2 x + (2\sin x \cos x)^2 = 2$$2\

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$x = (2n + 1)\frac{\pi}{4}$

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(B) Given equations are:$2\sin^2 x + \sin^2 2x = 2$ ... $(i)$$\sin 2x + \cos 2x = 	an x$ ... $(ii)$From $(i)$:$2\sin^2 x + (2\sin x \cos x)^2 = 2$$2\sin^2 x + 4\sin^2 x \cos^2 x = 2$$2\sin^2 x + 4\sin^2 x (1 - \sin^2 x) = 2$$2\sin^2 x + 4\sin^2 x - 4\sin^4 x = 2$$4\sin^4 x - 6\sin^2 x + 2 = 0$$2\sin^4 x - 3\sin^2 x + 1 = 0$$(2\sin^2 x - 1)(\sin^2 x - 1) = 0$So,$\sin^2 x = \frac{1}{2}$ or $\sin^2 x = 1$.If $\sin^2 x = 1$,then $\cos^2 x = 0$,which makes $	an x$ undefined.If $\sin^2 x = \frac{1}{2}$,then $\cos^2 x = \frac{1}{2}$,so $	an^2 x = 1$,which means $x = n\pi \pm \frac{\pi}{4}$.From $(ii)$:$\frac{2	an x}{1 + 	an^2 x} + \frac{1 - 	an^2 x}{1 + 	an^2 x} = 	an x$$2	an x + 1 - 	an^2 x = 	an x + 	an^3 x$$	an^3 x + 	an^2 x - 	an x - 1 = 0$$	an^2 x(	an x + 1) - 1(	an x + 1) = 0$$(	an^2 x - 1)(	an x + 1) = 0$So,$	an^2 x = 1$ or $	an x = -1$.Both imply $	an x = \pm 1$,which gives $x = n\pi \pm \frac{\pi}{4} = (2n + 1)\frac{\pi}{4}$.Thus,the common roots are $x = (2n + 1)\frac{\pi}{4}$.

List-$I$	List-$II$
$(A)$ $\sum \cot A$	$(i)$ $\frac{(a+b+c)^2}{4\Delta}$
$(B)$ $\sum \cot \frac{A}{2}$	$(ii)$ $\frac{a^2+b^2+c^2}{4\Delta}$
$(C)$ If $\tan A : \tan B : \tan C = 1 : 2 : 3$,then $\sin A : \sin B : \sin C =$	$(iii)$ $8 : 6 : 5$
$(D)$ If $\cot \frac{A}{2} : \cot \frac{B}{2} : \cot \frac{C}{2} = 3 : 7 : 9$,then $a : b : c =$	$(iv)$ $12 : 5 : 13$
	$(v)$ $\sqrt{5} : 2\sqrt{2} : 3$
	$(vi)$ $4\Delta$

The common roots of the equations $2\sin^2 x + \sin^2 2x = 2$ and $\sin 2x + \cos 2x = \tan x$ are

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