find the most general solutions for 2sin x+ 2cos x = 21 - (2)-1/2

Dear

2^{sin x} + 2^{cos x} = 2^{1 - (2)-1/2}

As we know AM ≥GM

                 (2^{sin x} + 2^{cos x})/2≥(2^{sin x}  2^{cos x} )^1/2    equality occur when 2^{sin x} = 2^{cos x}

       or   (2^{sin x} + 2^{cos x})/2≥(2^{sin x +cosx} )^1/2

  mimimum value occur sinx+cosx=-√2

      or   (2^{sin x} + 2^{cos x})/2≥(2^-√2 )^1/2

     or   (2^{sin x} + 2^{cos x})≥2^1-1/√2 

  so for solution equality should be occur

ie   2^{sin x} = 2^{cos x}

   sinx =cosx

 tanx =1

  x=n∏ +∏/4

Please feel free to post as many doubts on our discussion forum as you can.
If you find any question Difficult to understand - post it here and we will get you
the answer and detailed  solution very  quickly.

 We are all IITians and here to help you in your IIT JEE preparation.

All the best.
 
Regards,
Askiitians Experts
Badiuddin