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

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

    Hi!

    I need to establish this:Establish the Identity ((1+cosx+sinx)/(1+cosx-sinx)) = secx + tanx-untitled.png using the more basic trigonometric identities.

    I've tried all kinds of stuff for several hours without results. Any help would be highly appreciated!
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  2. #2
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    Re: Establish the Identity ((1+cosx+sinx)/(1+cosx-sinx)) = secx + tanx

    Quote Originally Posted by Benciticus View Post
    Hi!

    I need to establish this:Click image for larger version. 

Name:	Untitled.png 
Views:	4 
Size:	1.8 KB 
ID:	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!
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    Re: Establish the Identity ((1+cosx+sinx)/(1+cosx-sinx)) = secx + tanx

    Awesomesauce!!

    Thanks a bunch!
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    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.
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