Simplify the expression:

√[sin⁴x+4cos²x] - √[cos⁴x+4sin²x]

Source: AMC12 2002

√[sin(x)^4 + 4cos(x)^2] - √[cos(x)^4 + 4sin(x)^2]

s = sin(x)
c = cos(x)

√[s^4 + 4c^2] - √[c^4 + 4s^2]
√[s^4 + 4(1 - s^2)] - √[c^4 + 4(1 - c^2)]
√[s^4 - 4s^2 + 4] - √[c^4 - 4c^2 + 4]
√[s^2 - 2]^2 - √[c^2 - 2]^2
(s^2 - 2) - (c^2 - 2)
s^2 - c^2

All of the following are simplifications of the original.
(sin(x))^2 - (cos(x))^2
2*(sin(x))^2 - 1
cos(2x)  <-- the most simplified

Posted by Larry on 2023-04-16 08:11:53