Problem: When simplified 1+(x4−12x2)2\sqrt{1+\left(\dfrac{x^{4}-1}{2 x^{2}}\right)^{2}}1+(2x2x4−1)2 equals:
Answer Choices:
A. x4+2x2−12x2\dfrac{x^{4}+2 x^{2}-1}{2 x^{2}}2x2x4+2x2−1
B. x4−12x2\dfrac{x^{4}-1}{2 x^{2}}2x2x4−1
C. x2+12\dfrac{\sqrt{x^{2}+1}}{2}2x2+1
D. x22\dfrac{x^{2}}{\sqrt{2}}2x2
E. x22+12x2\dfrac{x^{2}}{2}+\dfrac{1}{2 x^{2}}2x2+2x21 Solution:
1+(x4−12x2)2=4x4+x8−2x4+14x4=(x4+12x2)2=x4+12x2=x22+12x2.\sqrt{1+\left(\dfrac{x^{4}-1}{2 x^{2}}\right)^{2}} =\sqrt{\dfrac{4 x^{4}+x^{8}-2 x^{4}+1}{4 x^{4}}}=\sqrt{\left(\dfrac{x^{4}+1}{2 x^{2}}\right)^{2}} =\dfrac{x^{4}+1}{2 x^{2}}=\dfrac{x^{2}}{2}+\dfrac{1}{2 x^{2}} . 1+(2x2x4−1)2=4x44x4+x8−2x4+1=(2x2x4+1)2=2x2x4+1=2x2+2x21.