General Solution Formula

1. sinx+cosx= √2
√2[sinx(1/√2)+cosx(1/√2)]= √2
√2sin(x+45˚)= √2
sin(x+45˚)=1
x+45=180n˚+(-1)n(90˚),where n is an integer.
x= 180n˚+(-1)n(90˚)-45˚

2.sin4x+sinx=0
2sin(5x/2)cos(3x/2)=0
sin(5x/2)=0 or cos(3x/2)=0
5x/2=180n˚ or 3x/2=360n˚90˚,where n is an integer.
x=72n˚ or x=240n˚60˚


3.2sinx sin3x=1
-cos4x+cos2x=1
-cos2(2x)+cos2x=1
-(2cos22x-1)+cos2x=1
2cos22x-cos2x=0
cos2x(2cos2x-1)=0
cos2x=0 or cos2x=1/2
2x=360n˚90˚ or 360n˚60˚,where n is an integer.
x=180n˚45˚ or 180n˚30˚


4.sinx=cos2x
sinx=1-2sin2x
2sin2x-sinx-1=0
(2sinx-1)(sinx+1)=0
sinx=1/2 or sinx=1
x=180n˚+(-1)n(30˚) or x=180n˚+(-1)n(90˚),where n is an integer.

5.cos2x-sin2x=1/2
cos2x=1/2
2x=360n˚60˚,where n is an integer.
x=180n˚30˚

6.
sinx+cosx= √2
√2[sinx(1/√2)+cosx(1/√2)]= √2
√2sin(x+45˚)= √2
sin(x+45˚)=1
x+45˚=180n˚+(-1)n(90˚),where n is an integer.
x= 180n˚+(-1)n(90˚)-45˚