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Math Help - A problem involving Inverse trignometry.

  1. #1
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    A problem involving Inverse trignometry.

    Solve the equation:
    Tan-1 [(x-1)/(x-2)] + Tan-1 [(x+1)/(x+2)] = π/4
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  2. #2
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    Re: A problem involving Inverse trignometry.

    We have
    \tan^{-1} \left( \frac{x-1}{x-2}\right)+\tan^{-1} \left( \frac{x+1}{x+2}\right)=\frac{\pi}{4}

    By using the famous identity, \arctan(x)+\arctan(y)= \arctan \left ( \frac{x+y}{1-xy}\right) we obtain

    \tan^{-1} \left( \frac{\frac{x-1}{x-2}+\frac{x+1}{x+2}}{1-\frac{x+1}{x+2} \frac{x-1}{x-2}}\right)=\frac{\pi}{4}

    \frac{(x-1)(x+2)+(x+1)(x-2)}{x^2-2-(x^2-1)} = \tan \left( \frac{\pi}{4}\right)

    x^2+2x-x-2+x^2-2x+x-2=-1

    2x^2-4=-1

    2x^2=3

    x= \pm \sqrt{\frac{3}{2}}
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  3. #3
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    Question Re: A problem involving Inverse trignometry.

    I had tried it the same way, but I had a few queries regarding it as thus.

    1)This identity is valid when the following conditions are satisfied:
    a) x>0
    b) y>0
    c) xy<1

    I was apprehensive about how I could ascertain the fact that the question satisfies these conditions. Or is it any other vital piece of concept I am lacking?

    2)Secondly could you please tell me if the site "www.wolframalpha.com", which is a computational knowledge engine, gives the same answer as the above(ie: root of 1.5), or first of all is it correct to tally this answer to the one that computational engine is producing?

    Well, I thank you sbhatnagar.

    And I would be glad if you, or anyone could help me with my doubts as stated above.
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