# Write the expressions in terms of cosine?

• Mar 11th 2009, 10:09 AM
mj.alawami
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 (Wondering))

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

Thank you
• Mar 11th 2009, 10:19 AM
e^(i*pi)
Quote:

Originally Posted by mj.alawami
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 (Wondering))

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

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

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From your expression you can say that $\displaystyle -1(sin^2(x)-1) = 1-sin^2(x) = cos^2(x)$
• Mar 11th 2009, 10:26 AM
mj.alawami
Quote:

Originally Posted by e^(i*pi)
$\displaystyle csc^2(x) = \frac{1}{sin^2(x)}$

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

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

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From your expression you can say that $\displaystyle -1(sin^2(x)-1) = 1-sin^2(x) = cos^2(x)$

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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
• Mar 11th 2009, 11:27 AM
e^(i*pi)
Quote:

Originally Posted by mj.alawami
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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 $\displaystyle sin^2(x)csc^2(x) = 1$