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Thread: Write the expressions in terms of cosine?

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    Write the expressions in terms of cosine?

    Q)Write the following expression in terms of cosine both in a simplified form?

    $\displaystyle \frac{1-csc^2x}{csc^2x} $

    Attempt- Is my attempt Correct )

    $\displaystyle -1(sin^2x-1) =-cos^2x $

    Thank you
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  2. #2
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    Quote Originally Posted by mj.alawami View Post
    Q)Write the following expression in terms of cosine both in a simplified form?

    $\displaystyle \frac{1-csc^2x}{csc^2x} $

    Attempt- Is my attempt Correct )

    $\displaystyle -1(sin^2x-1) =-cos^2x $

    Thank you
    $\displaystyle csc^2(x) = \frac{1}{sin^2(x)}$

    $\displaystyle
    1- \frac{1}{sin^2(x)} = \frac{sin^2(x) - 1}{sin^2(x)}$

    Overall the sum becomes:

    $\displaystyle \frac{sin^2(x) - 1}{sin^2(x)csc(x)}$ and so the denominator will be sin(x)

    $\displaystyle sin^2(x) - 1 = -cos^2(x) and sin(x) = \sqrt{1-cos^2(x)}$

    therefore we have $\displaystyle \frac{-cos^2(x)}{\sqrt{1-cos^2(x)}}$

    -----------

    From your expression you can say that $\displaystyle -1(sin^2(x)-1) = 1-sin^2(x) = cos^2(x)$
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    Quote Originally Posted by e^(i*pi) View Post
    $\displaystyle csc^2(x) = \frac{1}{sin^2(x)}$

    $\displaystyle
    1- \frac{1}{sin^2(x)} = \frac{sin^2(x) - 1}{sin^2(x)}$

    Overall the sum becomes:

    $\displaystyle \frac{sin^2(x) - 1}{sin^2(x)csc(x)}$ and so the denominator will be sin(x)

    $\displaystyle sin^2(x) - 1 = -cos^2(x) and sin(x) = \sqrt{1-cos^2(x)}$

    therefore we have $\displaystyle \frac{-cos^2(x)}{\sqrt{1-cos^2(x)}}$

    -----------

    From your expression you can say that $\displaystyle -1(sin^2(x)-1) = 1-sin^2(x) = cos^2(x)$
    -----------
    Your equation is correct but you can simplify the $\displaystyle csc^2$ in the denominator to $\displaystyle \frac{1}{sin^2} $ and then you can cancel the numerator and get $\displaystyle sin^2-1 $

    But i am not sure of my attempt

    Thank you
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    Quote Originally Posted by mj.alawami View Post
    -----------
    Your equation is correct but you can simplify the $\displaystyle csc^2$ in the denominator to $\displaystyle \frac{1}{sin^2} $ and then you can cancel the numerator and get $\displaystyle sin^2-1 $

    But i am not sure of my attempt

    Thank you
    My apologies I misread the question. Yes you can cancel the denominator because $\displaystyle sin^2(x)csc^2(x) = 1$
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