Going through solution and stuck on a step

[ridemx]

I am going through a given solution to a homework problem and have come to this statement

sina cosb/(sina cosb + cosa sinb) - cosa sinb/(sina cosb - cosa sinb)
= 1/(1 + tanb/tana) - 1/(1 + tana/tanb)

I have looked over every trig identity that I could find but cannot figure out how the Prof. got the second line from the first.

Thanks

 

[eumyang]

I am going through a given solution to a homework problem and have come to this statement

sina cosb/(sina cosb + cosa sinb) - cosa sinb/(sina cosb - cosa sinb)
= 1/(1 + tanb/tana) - 1/(1 + tana/tanb)

I have looked over every trig identity that I could find but cannot figure out how the Prof. got the second line from the first.

Thanks

Looks like he divided numerator & denominator in the 1st fraction by sin a cos b, and he divided numerator & denominator in the 2nd fraction by cos a sin b.

For the 1st fraction:
$$\frac{\sin a \cos b}{\sin a \cos b + \cos a \sin b}$$
$$\begin{aligned}<br /> =& \frac{\sin a \cos b}{\sin a \cos b + \cos a \sin b} \cdot \frac{\tfrac{1}{\sin a \cos b}}{\tfrac{1}{\sin a \cos b}}\\<br /> =& \frac{1}{1 + \tfrac{\cos a \sin b}{\sin a \cos b}}<br /> \end{aligned}$$

Can you figure it out from here?

 

[ridemx]

yep, that's it, just couldn't figure that out for some reason. Thanks a ton!