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Math Help - Verify the following identity...Need help

  1. #1
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    Verify the following identity...Need help

    Use x,y and r definitions of the Trig functions to verify the following identity.

    Tan^2 theta Csc theta+1/sin theta= Csc theta Sec^2 theta
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  2. #2
    No one in Particular VonNemo19's Avatar
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    Quote Originally Posted by Brndo4u View Post
    Use x,y and r definitions of the Trig functions to verify the following identity.

    Tan^2 theta Csc theta+1/sin theta= Csc theta Sec^2 theta
    So start by rewriting the left hand side,

    \overbrace{\left(\frac{y}{x}\right)^2}^{\tan^2\the  ta}\overbrace{\cdot\frac{r}{y}}^{\csc\theta}+\over  brace{\frac{r}{y}}^{\frac{1}{\sin\theta}}.

    Do the right hand side in a similar fashion, and then manipulate one of the sides so that it looks lik the other if necessary.
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  3. #3
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    Hello, Brndo4u!

    Use x,y and r definitions of the Trig functions to verify the following identity.

    . . \tan^2\theta\csc\theta + \frac{1}{\sin\theta} \:=\: \csc\theta\sec^2\theta

    The right side is: . \csc\theta\sec^2\theta \:=\:\left(\frac{r}{y}\right)\left(\frac{r}{x}\rig  ht)^2 \:=\:\frac{r^3}{x^2y}


    The left side is: . \tan^2\theta + \csc\theta + \frac{1}{\sin\theta} \;\;=\;\;\left(\frac{y}{x}\right)^2\left(\frac{r}{  y}\right) + \frac{1}{\left(\frac{y}{r}\right)}

    . . . =\;\;\frac{ry^2}{x^2y} + \frac{r}{y} \;=\;\frac{ry^2}{x^2y} + \frac{rx^2}{x^2y} \;\;=\;\;\frac{ry^2 + rx^2}{x^2y}

    . . . =\;\;\frac{r\overbrace{(x^2+y^2)}^{\text{This is }r^2}}{x^2y} \;\;=\;\;\frac{r\cdot r^2}{x^2y} \;\;=\;\;\frac{r^3}{x^2y}

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