the sum of all values of x belongs from 0 to 90 dedree satisfying sin2 2x+cos4 2x=3/4

x)² = 3/42 cos² 2x + (2sinx = 3/4     24 os 2x + c2sin

sin2 2x + (1 – sin² 2x)² = 3/4     sin2 2x + 1 – 2 sin² 2x + sin⁴ 2x  = 3/4

or  sin⁴ 2x – sin² 2x + 1/4 = 0    (sin² 2x – 1/2)² = 0

i.e. sin² 2x = 1/2  =  (1/)²   or   sin² 2x  =  sin² /4

  2x = n  /4  i.e.  x = n/2  /8

Put n=0, then x = /8  [ x belongs to 0 to /2 or 90]

Put n = 1, then x = /2  /8  =  5/8  or  3/8  [x = 5/8 isn’t possible for x belongs to 0 to /2]

Hence /8 and 3/8 are the required solutions whose

sum = /8 + 3/8 = /2 or 90 .

  = 3/4x2 sin⁴ + x2 sin²sin2 2x + 1 – 2   = 3/4    2x + (1 – sin² 2x)² 2sin= ¾ 2x + (cos² 2x)² 2sin

or  sin⁴ 2x – sin² 2x + 1/4 = 0    (sin² 2x – 1/2)² = 0

i.e. sin² 2x = 1/2  =  (1/)²   or   sin² 2x  =  sin² /4

  2x = n  /4  i.e.  x = n/2  /8

Put n=0, then x = /8  [ x belongs to 0 to /2 or 90]

Put n = 1, then x = /2  /8  =  5/8  or  3/8  [x = 5/8 isn’t possible for x belongs to 0 to /2]

Hence /8 and 3/8 are the required solutions whose

sum = /8 + 3/8 = /2 or 90 .