Trying to prove a trigonometric identity

Question

I've been trying to solve it for quite some time but I still don't get it why it is true.

The original equation is: \begin{equation*} 1-\frac{\sin{^2}\theta}{1-\cos\theta}=-\cos\theta. \end{equation*} My work so far: \begin{equation*} \frac{1-\cos\theta}{1-\cos\theta}-\frac{\sin{^2}\theta}{1-\cos\theta} =\frac{1-\cos\theta-\sin{^2}\theta}{1-\cos\theta} =\frac{(1-\sin{^2}\theta)-\cos\theta}{1-\cos\theta} \\ =\frac{\cos{^2}\theta-\cos\theta}{1-\cos\theta} =\frac{\cos\theta(\cos\theta-1)}{1-\cos\theta}. \end{equation*} I saw on some sites that this is equal to $-\cos\theta$ but I don't see why.

You've got $\cos(\theta)-1$ in the numerator which is $-1\cdot(1-\cos(\theta))$. Now cancel and you're done. — Michael Burr, May 14 '15 at 15:54

score 7 · Accepted Answer · answered May 14 '15 at 15:54

7

$\textbf{hint}$ $$ \cos\theta -1 = -(1-\cos\theta) $$

answered May 14 '15 at 15:54

Chinny84

14,186
2
22
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Got it. Thanks! – Dormin May 14 '15 at 17:20

Alan L · Answer 2 · 2015-05-22T10:02:21.840

1

You're very close. From where you left off

$\frac{\cos\theta(\cos\theta-1)}{1-\cos\theta}$

=$\frac{\cos\theta(-1)(1-\cos\theta)}{1-\cos\theta}$

=${\cos\theta(-1)}$

=${-\cos\theta}$

edited May 22 '15 at 10:02

answered May 14 '15 at 15:55

Alan L

13

Some equals signs would be nice. – N. F. Taussig May 15 '15 at 20:30

score 0 · Answer 3 · answered May 14 '15 at 15:58

0

Shorter: $$1-\frac{\sin{^2}\theta}{1-\cos\theta}=-\cos\theta\iff 1+\cos\theta=\frac{\sin{^2}\theta}{1-\cos\theta}\iff 1-\cos^2\theta=\sin^2\theta.$$

answered May 14 '15 at 15:58

Bernard

175,478

Trying to prove a trigonometric identity

3 Answers3