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Math Help - trig question (correct me) [part 1]

  1. #1
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    Smile trig question (correct me) [part 1]

    hello
    here's my function :
    let \mathbb{E} = [-\pi ,+\pi]
    x \in \mathbb{E} ; f(x) = \sqrt{\frac{1+cos(x)}{1-cos(x)}}
    and here's my questions :
    Prove that (\forall x \in \mathbb{D}) f(x) = \left |\tan (\frac{\pi}{2}-\frac{x}{2})  \right |
    Study the sign of \tan (\frac{\pi}{2}-\frac{x}{2}) in  ]0,+\pi]
    i proceeded like this :
    for the first one,we have \forall x\in \mathbb{R},1+cos(x) = 2cos^{2}(\frac{x}{2}) and 1-cos(x) = 2sin^{2}(\frac{x}{2})
    \Rightarrow \sqrt{\frac{1+cos(x)}{1-cos(x)}} =  \sqrt{\frac{2cos^{2} (\frac{x}{2})}{2sin^{2}(\frac{x}{2})}}=
    | \frac{cos(\frac{x}{2})}{sin(\frac{x}{2})}| = \left |\frac{1}{\tan (\frac{x}{2})}  \right | = \left |\tan (\frac{\pi}{2}-\frac{x}{2})  \right |
    for the second one,am i correct writing ; \forall x \in \mathbb{D},f(x) = tan(\frac{\pi}{2}-\frac{x}{2}) ( i already studied f )
    thanks for everyone
    by the way i'm trying to improve myself for writing proofs so if there is any small mistake please tell me
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  2. #2
    Senior Member nikhil's Avatar
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    Well....

    Quote Originally Posted by Raoh View Post

    Prove that (\forall x \in \mathbb{D}) f(x) = \left |\tan (\frac{\pi}{2}-\frac{x}{2})  \right |
    Study the sign of \tan (\frac{\pi}{2}-\frac{x}{2}) in  ]0,+\pi]
    well to me it seems that interval of D is not given and we are separetely suppose to check its sign in interval ]0,+pi]
    (you could have wrote - where D is given by ]0,+pi] )
    else seems correct to me
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  3. #3
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    Smile

    sorry about later,i forgot to mention the interval of \mathbb{D}.
    \mathbb{D} = [-\pi,0)\cup(0,+\pi]
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