question_answer

Consider the equation\[3\sqrt{{{[32{{\cos }^{6}}x-48{{\cos }^{4}}x+18\,{{\cos }^{2}}x-1]}^{2}}}=\sin \,x\]. Number of solutions of equation is

A)  \[9,\,\,x\in \,\,\left[ 0,\,\frac{\pi }{2} \right]\]   

B)  \[10,\,x\,\,\in \,\,[0,\,\,\pi ]\]

C)  \[8,\,\,x\,\in \,\,\left[ -\frac{\pi }{2},\,\frac{\pi }{2} \right]\]    

D)  \[12,\,\,x\,\,\in \,\,[-\pi ,\,\,\pi ]\]

Correct Answer: D

Solution :

 \[3\,\sqrt{{{[32\,{{\cos }^{6}}x-\,48\,{{\cos }^{4}}\,x\,+18\,{{\cos }^{2}}\,x-1]}^{2}}}\] \[=\sin \,x\] \[=3\,\sqrt{{{[2\,(16\,{{\cos }^{6}}x-24\,{{\cos }^{4}}x+9\,{{\cos }^{2}}\,x)-1]}^{2}}}\] \[=3\,\sqrt{2{{(4\,{{\cos }^{3}}x-3\,\cos \,x)}^{2}}-1}\] \[=3\,\sqrt{2\,{{\cos }^{2}}3x-1}\,\] \[3\,\sqrt{{{\cos }^{2}}\,6x}=\,\sin \,x\] \[3\,|\cos \,6x|\,=\,\sin \,x\] \[6,\,x\in \,\left[ 0,\,\frac{\pi }{2} \right]\] \[12,\,\,x\,\in \,[0,\pi ]\] Clearly, number of solutions, \[=\,6,\,x\in \,\left[ -\frac{\pi }{2},\,\frac{\pi }{2} \right]\]  \[=12,\,x\,\in \,[-\pi ,\,\pi ]\]