1. ## simplifying the expression

I'm having trouble with this

$\displaystyle (1-sin^2(x)) / sin(x) - csc(x)$

2. Originally Posted by yeunju
I'm having trouble with this

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

Can you continue?

3. $\displaystyle \frac{1-\sin^2x}{\sin x-\csc x}=\frac{1-\sin^2x}{\sin x-\frac{1}{\sin x}}=\frac{1-\sin^2x}{\frac{\sin^2x-1}{\sin x}}=-\sin x$

4. Originally Posted by yeunju
I'm having trouble with this

$\displaystyle (1-sin^2(x)) / sin(x) - csc(x)$
$\displaystyle \color{red} sin^2(x) + cos^2(x) = 1 \qquad \implies sin^2(x) - 1 = -cos^2(x)$

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

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

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

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

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

Denominator = $\displaystyle sin(x) - csc(x) = sin(x) - \frac{1}{sin(x)}$

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

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

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

So fraction is

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

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

$\displaystyle =~ -sin(x)$