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Math Help - Using calculus to deduce a trigonometric identity.

  1. #1
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    Using calculus to deduce a trigonometric identity.

    g(x)=tan^{-1} x+ tan^{-1} (\frac{1}{x}). Find g'(x) and hence deduce that tan^{-1} x + tan^{-1} (\frac{1}{x}) = \frac{\pi}{2}
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  2. #2
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    Quote Originally Posted by nerdzor View Post
    g(x)=tan^{-1} x+ tan^{-1} (\frac{1}{x}). Find g'(x) and hence deduce that tan^{-1} x + tan^{-1} (\frac{1}{x}) = \frac{\pi}{2}
    Can you get g'(x)? If not, where do you get stuck? If you can get it, please post the answer you got (and show your working).
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  3. #3
    Senior Member pankaj's Avatar
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     <br />
g(x)=tan^{-1} x+ tan^{-1} (\frac{1}{x})<br />

     <br />
g'(x)=\frac{1}{1+x^2}-\frac{1}{1+\frac{1}{x^2}}(\frac{-1}{x^2})=0<br />

    Since g'(x)=0,therefore g(x) must be a constant.

     <br />
g(x)=g(1)=tan^{-1}1+tan^{-1}1=\frac{\pi}{2}<br />

    Now,you should have mentioned that x>0.

    Otherwise if x<0 then g(x)=-\frac{\pi}{2}
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