Question

cosx+sinx=√2 sin3x=?

Answer 1

Cos x + Sin x = √2
square on both sides:
Cos² x + Sin² x + 2 Sin x  Cos x = 2
     =>  Sin 2x = 1
     => Cos 2x = √[1 - Sin² 2x ] = 0

(Cos x - Sin x)² = Cos² x + Sin² x - sin 2x = 1  - 1  = 0
  => Cos x = Sin x
   => Cos² x = Sin² x = 1/2
     =>  Cos x = Sin x = 1/√2

Sin 3 x = Sin x Cos 2x + Cos x Sin 2x
         =  0 + 1/√2 * 1   = 1/√2

OR, alternately,
     we know Sin 2x = 1    => x = π/4
       => Sin 3x = Sin 3π/4 =1/√2