• # question_answer The real roots of the equation, ${{\cos }^{7}}x+{{\sin }^{4}}x=1$ in the interval $(-\pi ,\,\,\pi )$ A) $\frac{-\pi }{2},\,0$               B)  $\frac{-\pi }{2},\,0\,\frac{\pi }{2}$ C)  $\frac{\pi }{2},\,0$                D)  $0,\frac{\pi }{2},\,\frac{\pi }{2}$

$\because$    ${{\cos }^{7}}x\le {{\cos }^{2}}x$ and      ${{\sin }^{4}}x\le \,{{\sin }^{2}}x$ On adding Eqs. (i) and (ii), we get $\Rightarrow$            ${{\cos }^{7}}x+{{\sin }^{4}}x\le 1$ But given, ${{\cos }^{7}}x+{{\sin }^{4}}x=1$ Equality holds only if ${{\cos }^{7}}x={{\cos }^{2}}x$ and      ${{\sin }^{4}}x={{\sin }^{2}}x$ Both are satisfies by $x=\pm \,\frac{\pi }{2},\,0$