Solve 1/sqrt{2}sinx+1/sqrt{2}cosx

$$\frac { 1 } { \sqrt { 2 } } \sin x + \frac { 1 } { \sqrt { 2 } } \cos x$$

Factor

$$\sin(\frac{4x+\pi }{4})$$

Solution Steps

Rationalize the denominator of \frac{1}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$factor(\frac{\sqrt{2}}{\left(\sqrt{2}\right)^{2}}\sin(x)+\frac{1}{\sqrt{2}}\cos(x))$$

The square of \sqrt{2} is 2.

$$factor(\frac{\sqrt{2}}{2}\sin(x)+\frac{1}{\sqrt{2}}\cos(x))$$

Express \frac{\sqrt{2}}{2}\sin(x) as a single fraction.

$$factor(\frac{\sqrt{2}\sin(x)}{2}+\frac{1}{\sqrt{2}}\cos(x))$$

Rationalize the denominator of \frac{1}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$factor(\frac{\sqrt{2}\sin(x)}{2}+\frac{\sqrt{2}}{\left(\sqrt{2}\right)^{2}}\cos(x))$$

The square of \sqrt{2} is 2.

$$factor(\frac{\sqrt{2}\sin(x)}{2}+\frac{\sqrt{2}}{2}\cos(x))$$

Express \frac{\sqrt{2}}{2}\cos(x) as a single fraction.

$$factor(\frac{\sqrt{2}\sin(x)}{2}+\frac{\sqrt{2}\cos(x)}{2})$$

Since \frac{\sqrt{2}\sin(x)}{2} and \frac{\sqrt{2}\cos(x)}{2} have the same denominator, add them by adding their numerators.

$$factor(\frac{\sqrt{2}\sin(x)+\sqrt{2}\cos(x)}{2})$$

Consider \sqrt{2}\left(\sin(x)+\cos(x)\right). Factor out \sqrt{2}.

$$\sqrt{2}\left(\sin(x)+\cos(x)\right)$$

Rewrite the complete factored expression.

$$\frac{\left(\sin(x)+\cos(x)\right)\sqrt{2}}{2}$$

Evaluate

$$\sin(\frac{4x+\pi }{4})$$