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Math Help - Prove...

  1. #1
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    Question Prove...

    Prove
    tan70=tan 20 + 2tan 50...
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  2. #2
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    Quote Originally Posted by amitangshu94 View Post
    Prove
    tan70=tan 20 + 2tan 50...
    A geometric proof can be achieved by drawing a sketch.

    Draw a right-angled triangle, base=x, vertical side=y, hypotenuse=w.
    The angle at the lower left corner is 50 degrees, 90 degrees at the lower right,
    top angle=40 degrees.

    Draw another hypotenuse at 70 degrees from the lower left apex and continue the vertical,
    as shown in the attachment.


    \displaystyle\ tan20^o=\frac{x}{k+y}=\frac{x}{w+y}

    since w = k, as the top triangle is isosceles.

    \displaystyle\ tan70^0=\frac{k+y}{x}=\frac{w+y}{x}=\frac{1}{tan20  ^o}

    \displaystyle\ tan50^o=\frac{y}{x}\Rightarrow\ 2tan50^o=\frac{2y}{x}

    \displaystyle\ 2tan50^o+tan20^o=\frac{2y}{x}+\frac{x}{w+y}=\frac{  2yw+2y^2+x^2}{x(w+y)}

    =\displaystyle\frac{2yw+y^2+y^2+x^2}{x(w+y)}

    From Pythagoras' theorem....

    x^2+y^2=w^2

    therefore....

    \displaystyle\ 2tan50^o+tan20^o=\frac{w^2+2yw+y^2}{x(w+y)}=\frac{  (w+y)^2}{x(w+y)}=\frac{w+y}{x}

    which is tan70^o
    Attached Thumbnails Attached Thumbnails Prove...-tan-angles-.jpg  
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