Clear of fractions by multiplying thru by 5



Use identity 
              



Isolate the term with the square root



Square both sides:







divide through by 2



Factor the right side



Use the zero-factor property

  

   

   

Since we are given 



which means x is in QII where the sine is positive,
we can eliminate the negative value for the sine.



Use identity
              













Edwin

It can be done in much simpler manner.


Start from the given equation

    sin(x) + cos(x) = .


Square both sides

    sin^2(x) + 2*sin(x)*cos(x) + cos^2(x) = .


Replace  sin^x) + cos^2(x) by 1

    1 + 2*sin(x)*cos(x) = 


Replace 2*sin(x)*cos(x) by sin(2x)

    sin(2x) =  - 1,   or   sin(2x) = .


Hence, cos^2(2x) =  =  =  =  = .


Since the angle (2x) is in QIV, cos(2x) is positive.


Hence,  cos(2x) = positive square root of  =  = .


The proof is completed.