How to simplify $\frac{\frac{\cos x}{2\sqrt{x}} + \sqrt{x}\sin x}{\cos^2x}$, step by step?

Question

I want to simplify this expression

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

and I know this is the answer

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

How can I get there step by step? Thanks in advance!

Answer 1

$$\frac{\frac{\cos(x)}{2\sqrt{x}} + \sqrt{x}\sin(x)}{\cos^2(x)}$$ The LCM of the numerator terms is $\space 2\sqrt{x}\space$ so the expression becomes $$\dfrac{\frac{\cos(x)}{2\sqrt{x}} + \frac{2x\sin(x)}{2\sqrt{X}}}{\cos^2(x)}$$ A denominator in a numerator-fraction is part of the denominator of the whole fraction

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

Answer 2

Well, we have:

$$\frac{\frac{\cos(x)}{2\sqrt{x}}+\sqrt{x}\sin(x)}{\cos^{2}(x)} =\frac{\frac{\cos(x)}{2\sqrt{x}}+\frac{2\sqrt{x}}{2\sqrt{x}}\sqrt{x}\sin(x)}{\cos^{2}(x)}=\frac{\frac{\cos(x)+2x\sin(x)}{2\sqrt{x}}}{\cos^{2}(x)}=\boxed{\frac{\cos(x)+2x\sin(x)}{2\sqrt{x}\cos^{2}(x)}}$$