1. ## Simplifying

I'm having difficulties simplifying the expression.

$\displaystyle cos(\frac{pi}2-x)csc(-x)$

2. Originally Posted by Altair_xI
I'm having difficulties simplifying the expression.

$\displaystyle cos(\frac{pi}2-x)csc(-x)$
Remember that $\displaystyle \cos \bigg(\frac{\pi}{2} - \theta\bigg) = \sin(\theta)$

And $\displaystyle \csc(\theta) = \frac{1}{\sin(\theta)}$

Hence:

$\displaystyle \cos \bigg(\frac{\pi}{2} - x \bigg) \csc(-x) = \sin(x) \frac{1}{\sin(-x)} = \sin(x) \frac{1}{-\sin(x)} = \frac{\sin(x)}{-\sin(x)} = -1$

3. Originally Posted by Altair_xI
I'm having difficulties simplifying the expression.

$\displaystyle cos(\frac{pi}2-x)csc(-x)$
Note that $\displaystyle \cos(a-b)=\cos(a)\cos(b)+\sin(a)\sin(b)$ and

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

$\displaystyle \sin(-x)=-\sin(x)$ sine is an odd function

$\displaystyle \cos\left( \frac{\pi}{2}-x\right)\csc(-x)=\frac{\cos\left( \frac{\pi}{2}-x\right)}{\sin(-x)}=$

$\displaystyle \frac{\cos(\frac{\pi}{2})\cos(x)+\sin(\frac{\pi}{2 })\sin(x) }{-\sin(x)}= \frac{\sin(x)}{-\sin(x)}=-1$

4. Thank you. I see where I've messed up now. Many thanks.