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Math Help - changing expression

  1. #1
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    changing expression

    Hi,

    I was just wondering if you have an expression like this  sec^{4} \theta - tan^{4} \theta and if I take the sqaure root of this expression would it end up looking like this ;  sec^{2} \theta - tan^{2} \theta ?

    thanks!
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  2. #2
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    Quote Originally Posted by Tweety View Post
    Hi,

    I was just wondering if you have an expression like this  sec^{4} \theta - tan^{4} \theta and if I take the sqaure root of this expression would it end up looking like this ;  sec^{2} \theta - tan^{2} \theta ?

    thanks!
    in general, no. \sqrt{a^4 - b^4} \ne a^2 - b^2

    but you can do this ....

    \sec^4{t} - \tan^4{t} =

    (\sec^2{t} - \tan^2{t})(\sec^2{t} + \tan^2{t}) =

    [(1 + \tan^2{t}) - \tan^2{t}](\sec^2{t} + \tan^2{t}) =

    (1)(\sec^2{t} + \tan^2{t}) = \sec^2{t}+\tan^2{t} = 1 + 2\tan^2{t}
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  3. #3
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    Quote Originally Posted by skeeter View Post
    in general, no. \sqrt{a^4 - b^4} \ne a^2 - b^2

    but you can do this ....

    \sec^4{t} - \tan^4{t} =

    (\sec^2{t} - \tan^2{t})(\sec^2{t} + \tan^2{t}) =

    [(1 + \tan^2{t}) - \tan^2{t}](\sec^2{t} + \tan^2{t}) =

    (1)(\sec^2{t} + \tan^2{t}) = \sec^2{t}+\tan^2{t} = 1 + 2\tan^2{t}
    Okay thank you. Also how do you know that this expression is a difference of two squares?

    or is it in general any expression like that can be expressed as a difference of two squares?
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  4. #4
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    Quote Originally Posted by Tweety View Post
    Okay thank you. Also how do you know that this expression is a difference of two squares?

    or is it in general any expression like that can be expressed as a difference of two squares?
    If there are only two terms and both are squares it's likely to be the difference of two squares. Essentially keep a lookout for quadratics with no x term.

    a^n - b^n = (a^{\frac{n}{2}} - b^{\frac{n}{2}})(a^{\frac{n}{2}} + b^{\frac{n}{2}})

    This reduces to an integer for even values of n

    In the world of trig powers sin^n(x) = (sin(x))^n \: , \: n \in \mathbb{Z^+}
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