Can somebody help to prove that:
$$\frac{\cos a - \sin a + 1}{\cos a + \sin a - 1 } = \frac{\sin a}{1-\cos a}$$
Asked
Active
Viewed 104 times
-1
Blue
- 75,673
-
3Please show some efforts first. – Ѕᴀᴀᴅ Feb 05 '18 at 02:54
-
Related: "Better proof for $\frac{1+\cos x + \sin x}{1 - \cos x + \sin x} \equiv \frac{1+\cos x}{\sin x}$". – Blue Feb 05 '18 at 02:56
-
Multiply by both denominators and cancel out terms. There would be left $1=\sin^2 a+\cos^2 a$. – user_194421 Feb 05 '18 at 03:01
-
Multiply thru by the product of the denominators and simplify. I.e. if $B\ne 0\ne D$ then $A/B=C/D\iff AD=BC.$ – DanielWainfleet Feb 05 '18 at 03:02
-
As requested by Alex, it's generally best for questioners to include a sense of what they know about a problem and where they got stuck, so that potential answerers can target their responses to an appropriate skill level, and without wasting time telling the questioner things that are already known. Posting a problem without this kind of context looks like you're just trying to get people to do your homework for you, often resulting in down-votes and/or votes-to-close. – Blue Feb 05 '18 at 03:04
1 Answers
-1
\begin{eqnarray*} \frac{ \cos a -\sin a +1}{\cos a +\sin a -1} \times \frac{1-\cos a}{1-\cos a} =\frac{\color{red}{\cos a}-\sin a+\overbrace{1-\cos^2 a}^{\sin^2 a}+\sin a \cos a\color{red}{-\cos a}}{(\cos a +\sin a -1)((1-\cos a)} =\frac{\sin a (\cos a +\sin a -1)}{(1-\cos a)(\cos a +\sin a -1)} \end{eqnarray*}
Donald Splutterwit
- 36,613
- 2
- 26
- 73