Answer:
Apply the angle sum identity [tex]\sin(a + b) = \sin(a)\, \cos(b) + \cos(a)\, \sin(b)[/tex].
Step-by-step explanation:
By the angle sum identity, [tex]\sin(a + b) = \sin(a)\, \cos(b) + \cos(a)\, \sin(b)[/tex] for any angles [tex]a[/tex] and [tex]b[/tex].
Apply this identity to rewrite the left-hand side of the equation:
[tex]\begin{aligned}& \sin\left(\frac{\pi}{4} + x\right) \\ =\; & \sin\left(\frac{\pi}{4}\right)\, \cos(x) + \cos\left(\frac{\pi}{4}\right)\, \sin(x) \end{aligned}[/tex].
The angle [tex](\pi/4)[/tex] is equivalent to [tex]45^{\circ}[/tex], which corresponds to the isosceles right triangle: [tex]\sin(\pi/4) = \sqrt{2} / 2[/tex], [tex]\cos(\pi/4) = \sqrt{2} / 2[/tex].
Substitute these two values into the expression above and simplify:
[tex]\begin{aligned}& \sin\left(\frac{\pi}{4} + x\right) \\ =\; & \sin\left(\frac{\pi}{4}\right)\, \cos(x) + \cos\left(\frac{\pi}{4}\right)\, \sin(x) \\ =\; & \frac{\sqrt{2}}{2}\, \cos(x) + \frac{\sqrt{2}}{2}\, \sin(x) \\ =\; & \frac{\sqrt{2}}{2} \, (\sin(x) + \cos(x))\end{aligned}[/tex].
Thus, [tex]\sin((\pi/4) + x) = (\sqrt{2} / 2)\, (\sin(x) + \cos(x))[/tex] as requested.
