# verifying idenitity

• May 6th 2009, 07:51 PM
wolfxpax
verifying idenitity
how would i verify this identity?
(1+sec(-x))/(sin(-X)+tan(-x))= -csc x
• May 6th 2009, 10:25 PM
Hello wolfxpax
Quote:

Originally Posted by wolfxpax
how would i verify this identity?
(1+sec(-x))/(sin(-X)+tan(-x))= -csc x

First you need to know how to handle the negative signs:

• $\displaystyle \sin(-x) = -\sin x$
• $\displaystyle \cos(-x) = \cos x$

So it follows that $\displaystyle \tan(-x) =-\tan x, \sec(-x) = \sec x$, and so on.

So $\displaystyle \frac{1 +\sec(-x)}{\sin(-x)+\tan(-x)} =\frac{1 +\sec(x)}{-\sin(x)-\tan(x)}$

$\displaystyle = \frac{1+\sec x}{-\sin x - \dfrac{\sin x}{\cos x}}$

$\displaystyle = \frac{1+\sec x}{-\sin x - \dfrac{\sin x}{\cos x}}\times\color{red}\frac{\cos x}{\cos x}$

$\displaystyle = \frac{\cos x + 1}{-\sin x\cos x - \sin x}$

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

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

$\displaystyle =-\csc x$

• May 7th 2009, 02:06 PM
Soroban
Hello, wolfxpax!

A slightly different approach . . .

Quote:

$\displaystyle \frac{1+\sec(\text{-}x)}{\sin(\text{-}x)+\tan(\text{-}x)} \:=\: -\csc x$
As Grandad pointed out: .$\displaystyle \begin{array}{ccc} \sec(\text{-}x) &=& \sec(x) \\ \sin(\text{-}x) &=& \text{-}\sin(x) \\ \tan(\text{-}x) &=& \text{-}\tan(x) \end{array}$

So we have: .$\displaystyle \frac{1+\sec x}{\text{-}\sin x - \tan x} \;=\;\frac{1+\sec x}{\text{-}\sin x - \frac{\sin x}{\cos x}} \;=\;\frac{1+\sec x}{\text{-}\sin x\left(1 + \frac{1}{\cos x}\right)}$

. . . . . . . $\displaystyle = \;\frac{1 + \sec x}{\text{-}\sin x(1 + \sec x)} \;=\;\frac{1}{-\sin x} \;=\;-\csc x$