# Establish the Identity ((1+cosx+sinx)/(1+cosx-sinx)) = secx + tanx

• April 29th 2012, 04:25 PM
Benciticus
Establish the Identity ((1+cosx+sinx)/(1+cosx-sinx)) = secx + tanx
Hi!

I need to establish this:Attachment 23731 using the more basic trigonometric identities.

I've tried all kinds of stuff for several hours without results. Any help would be highly appreciated!
• April 29th 2012, 05:21 PM
skeeter
Re: Establish the Identity ((1+cosx+sinx)/(1+cosx-sinx)) = secx + tanx
Quote:

Originally Posted by Benciticus
Hi!

I need to establish this:Attachment 23731 using the more basic trigonometric identities.

I've tried all kinds of stuff for several hours without results. Any help would be highly appreciated!

maybe there is an easier way ???

$\frac{(1+\cos{x})+\sin{x}}{(1+\cos{x})-\sin{x}} \cdot \frac{(1+\cos{x})+\sin{x}}{(1+\cos{x})+\sin{x}}$

$\frac{(1+\cos{x})^2 + 2(1+\cos{x})\sin{x} + \sin^2{x}}{(1+\cos{x})^2 - \sin^2{x}}$

$\frac{1+2\cos{x}+\cos^2{x}+2\sin{x}+2\cos{x}\sin{x }+\sin^2{x}}{1+2\cos{x}+\cos^2{x}-\sin^2{x}}$

$\frac{2+2\cos{x}+2\sin{x}+2\cos{x}\sin{x}}{2\cos{x }+2\cos^2{x}}$

$\frac{2(1+\cos{x})+2\sin{x}(1+\cos{x})}{2\cos{x}(1 +\cos{x})}$

$\frac{2(1+\cos{x})[1+\sin{x}]}{2(1+\cos{x})\cos{x}}$

$\frac{1+\sin{x}}{\cos{x}}$

$\frac{1}{\cos{x}} + \frac{\sin{x}}{\cos{x}}$

$\sec{x} + \tan{x}$

... what a pain-in-the-@ identity!
• April 30th 2012, 09:15 AM
Benciticus
Re: Establish the Identity ((1+cosx+sinx)/(1+cosx-sinx)) = secx + tanx
Awesomesauce!!

Thanks a bunch!
• April 30th 2012, 09:55 AM
biffboy
Re: Establish the Identity ((1+cosx+sinx)/(1+cosx-sinx)) = secx + tanx
Alternatively write the right hand side as 1/cosx+sinx/cosx= (1+sinx)/cosx and then show that LHS-RHS=0 by putting on common denominator.