Find the maximum and minimum values of sin^4x + cos^2x and hence or otherwise find the maximum value of sin^1000 x + cox^2008 x

sin4x + cos2x  = (1-cos2x )2 + cos2
= 1 + cos4x -2cos2x + cos2x
= 1+ cos4x -cos2x
= 1/4  + 3/4  + cos4x - cos2x
=(cos2x -12)2 +34 =(2cos2x -12)2 +34=(cos2x)24+34 As (cos2x)2  will lies between [0,1]So above expression has minimum value when  (cos2x)2  = 0 And maximum value when (cos2x)2 = 1 Hence the range is [34,1]

And sin1000x  + cos2008x

Its maximum value will be 1 , as both sinx and cosx range are less than equal to one , so if a number less than 1 has exponents very high like 1000 or 2008 , then it will eventually lead to very small value and ultimately zero.
So the smallest value will be zero and maximum value will  be 1 , when x  = 0 or 90 degree .

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