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Thread: Equating two equations - basic algebra questions re exponents and trig functions

  1. #1
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    Equating two equations - basic algebra questions re exponents and trig functions

    Hi,

    I have the following:
    Consider the following two expressions:

    f(x) = e ^(2x) − (e^(x) cos x)^ (2)/x^2 and g(x) =  (e^xsin x/x)^2

    I have to show that these equations are equal for x not equal to 0.

    I have a couple of basic questions:

    When an expression is expressed to the power 2 (say) is it that case that to multiply a term by an exponent you add exponents?

    I am given the hint: cos^(2) x + sin^(2) x (cos squared x plus sin squared x) = 1

    I have fiddled around a bit however can't work out how that hint helps...does anyone have any other ideas?

    Any help welcome

    Kind regards
    Beetle
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  2. #2
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    Re: Equating two equations - basic algebra questions re exponents and trig functions

    When an expression is expressed to the power 2 (say) is it that case that to multiply a term by an exponent you add exponents?
    you don't "multiply" a term by an exponent ... the two basic rules are $(a^x)^y = a^{xy}$ and $a^x \cdot a^y = a^{x+y}$


    Consider the following two expressions:

    f(x) = [e ^(2x) − (e^(x) cos x)^ (2)]/x^2 and g(x) = (e^xsin x/x)^2

    I have to show that these equations are equal for x not equal to 0.
    ok ... I'm gonna make a request that you start using brackets to make your expressions clear

    as you've written it ... $f(x) = e^{2x} - \dfrac{(e^x\cos{x})^2}{x^2}$

    what it really is ...

    $f(x) = \dfrac{e^{2x}-(e^x\cos{x})^2}{x^2}$

    proceeding ...

    $f(x) = \dfrac{e^{2x}-e^{2x}\cos^2{x}}{x^2}$

    $f(x) = \dfrac{e^{2x}(1-\cos^2{x})}{x^2}$

    can you finish?
    Last edited by topsquark; Apr 2nd 2016 at 09:59 AM.
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  3. #3
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    Re: Equating two equations - basic algebra questions re exponents and trig functions

    Hi,

    First of all my apologies for the bracket problem. I actually had more brackets and then following feedback last week took some of them out – to many it seems. I tried typing my posts in work and then copying and pasting however that still doesn't return the correct syntax - it reverts to what is used in this forum.

    Thank you for your help – there is one step in your working I don’t understand that I have pointed out – following that I can get from your last line of working to the solution.

    Would you be able to explain the missing step?

    f(x)=e^(2x)-(e^(2x) cos(^2) x)/x(^2)

    You gave me the expression with (e^(x)cos(^2)) multiplied out right?

    So f(x) = e^(2x) - e^(2x) cos^(2)x/x^(2)

    I don’t understand the next step:

    Being f(x)=e^(2x)(1-cos^(2)x)/x^(2)

    Are you able to explain how you got this step. Thank you again.

    Kind regards
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  4. #4
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    Re: Equating two equations - basic algebra questions re exponents and trig functions

    The numerator is like:

    a - ab = a(1-b)

    where a=e^{2x} is a common factor and b=cos^2x.
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