general solution~

1)cos x = sec x

cos x = 1/cos x
cos2 x = 1
cosx = ±1
x = 180no

2)2sin2x﹣5cosx﹣4 = 0

2(1﹣cos2x)﹣5cosx﹣4 = 0
2﹣2cos2x﹣5cosx﹣4 = 0
2cos2x + 5cosx + 2 = 0
(2cosx + 1)(cosx + 2) = 0
cosx = -1/2 或 cosx = -2 (捨去)
x = 360no±120o

3)sinx + cosx = √2
√2 cos(θ﹣45o) = √2
cos(θ﹣45o) = 1
θ﹣45o = 360no
θ = 360no+45o

4)sin4x + sinx = 0
2sin[(4x+x)/2] cos[(4x﹣x)/2] = 0
2sin(5x/2) cos(3x/2) = 0
sin(5x/2) = 0 或 cos(3x/2) = 0
5x/2 = 180no 或 3x/2 = 360no±90o
x = 72no 或 x = 240no±60o

5)2sinx sin3x = 1
2{-1/2[cos(x+3x)-cos(x-3x)]}=1

-(cos4x)+cos(2x)=1

-(2cos22x﹣1)+cos 2x = 1

-2cos2x + cos 2x = 0

cos2x(-2cos2x + 1) = 0

cos 2x = 0 or (-2cos2x + 1) = 0

cos 2x = cos 90o or cos2x = -1/2

2x=360no±90o or cos2x = cos 120o

x=180no±45o or 2x=360no±120o

x=180no±45o or x=180no±60o