# Proving $\frac{\cos a - \sin a + 1}{\cos a + \sin a - 1 } = \frac{\sin a}{1-\cos a}$ [duplicate]

Can somebody help to prove that:

$$\frac{\cos a - \sin a + 1}{\cos a + \sin a - 1 } = \frac{\sin a}{1-\cos a}$$

## marked as duplicate by Carl Mummert, lab bhattacharjee trigonometry StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Feb 5 '18 at 4:28

• Multiply by both denominators and cancel out terms. There would be left $1=\sin^2 a+\cos^2 a$. – user_194421 Feb 5 '18 at 3:01
• 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 5 '18 at 3:02