Sum of roots of the equation x4−2x2sin2⁡πx2+1=0 is

x4−2x2sin2⁡π2x+1=0⇒ x4+1=2x2sin2⁡πx2⇒ x4+1=2x21−cos2⁡πx2⇒ x2−12+2x2cos2⁡πx2=0⇒ x2−1=0 and 2x2cos2⁡πx2=0⇒ x=±1

Sum of roots of the equation x4−2x2sin2⁡πx2+1=0 is

x4−2x2sin2⁡π2x+1=0⇒ x4+1=2x2sin2⁡πx2⇒ x4+1=2x21−cos2⁡πx2⇒ x2−12+2x2cos2⁡πx2=0⇒ x2−1=0 and 2x2cos2⁡πx2=0⇒ x=±1

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Sum of roots of the equation $x^{4} - 2 x^{2} \sin^{2} \frac{πx}{2} + 1 = 0$ is

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Detailed Solution

$\begin{array}{l} x^{4} - 2 x^{2} \sin^{2} \frac{π}{2} x + 1 = 0 \\ \Rightarrow x^{4} + 1 = 2 x^{2} \sin^{2} \frac{πx}{2} \\ \Rightarrow x^{4} + 1 = 2 x^{2} (1 - \cos^{2} \frac{πx}{2}) \\ \Rightarrow {(x^{2} - 1)}^{2} + 2 x^{2} \cos^{2} \frac{πx}{2} = 0 \\ \Rightarrow (x^{2} - 1) = 0 and 2 x^{2} \cos^{2} \frac{πx}{2} = 0 \\ \Rightarrow x = \pm 1 \end{array}$