For some reason I CANT get the answer for

(1-sin^2X)/(sinX-cscX)= -sinX

2. Originally Posted by amanda0603
For some reason I CANT get the answer for

(1-sin^2X)/(sinX-cscX)= -sinX

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

set the same denominator:

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

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

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

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

which cancels to $\displaystyle -sin(x)$

3. well, $\displaystyle sin^2(x)+cos^2(x)=1$, so $\displaystyle 1-sin^2(x)=$$\displaystyle cos^2(x)$

4. Hello, Amanda!

Another approach . . .

$\displaystyle \frac{1-\sin^2\!x}{\sin x-\csc x} \:=\: -\sin x$

$\displaystyle \text{We have: }\;\frac{1-\sin^2\!x}{\sin x-\csc x} \:=\:\frac{1-\sin^2\!x}{\sin x - \frac{1}{\sin x}}$

Multiply by $\displaystyle \frac{\sin x}{\sin x}\!:\;\;\frac{\sin x}{\sin x}\cdot\frac{1-\sin^2\!x}{\sin x - \frac{1}{\sin x}} \;=\;\frac{\sin x(1 - \sin^2\!x)}{\sin^2\!x - 1}$

Factor and reduce: .$\displaystyle \frac{-\sin x\,({\color{red}\rlap{////////}}\sin^2\!x-1)}{{\color{red}\rlap{////////}}\sin^2\!x-1} \;=\; -\sin x$