If tan x = 1, evaluate sin x + cos x, leaving your answer in the surd form

If tan x = 1, evaluate sin x + cos x, leaving your answer in the surd form

Answer Details

If \(\tan x = 1\), then we can find the values of sin x and cos x using the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\). Since \(\tan x = \frac{\sin x}{\cos x} = 1\), we can say that \(\sin x = \cos x\). So we have, \[\sin^2 x + \cos^2 x = 2\cos^2 x = 1 \implies \cos x = \pm\frac{1}{\sqrt{2}}\] Using \(\sin x = \cos x\), we can say that \(\sin x = \pm\frac{1}{\sqrt{2}}\) Now, \[\sin x + \cos x = \pm\frac{1}{\sqrt{2}} \pm \frac{1}{\sqrt{2}} = \pm\sqrt{2}\] Since \(\tan x = 1\), x must be in the first quadrant, where both \(\sin x\) and \(\cos x\) are positive. Therefore, we have \[\sin x + \cos x = \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} = \sqrt{2}\] Hence, the answer is \(\sqrt{2}\). Therefore, the option that represents this answer is: $\sqrt{2}$.