Expert: Socrates - 9/26/2006

Question
Hi,


I am trying to differentiate the following.

y = ( cos x + sin x ) / (cos x - sin x)

I have to show that the answer is    
2 / ( 1 - sin x)


I used the quotient rule and got as far as this

[ (cos x - sin x).(-sin x + cos x) -
 (cos x + sin).(-sin x - cos x) ] /
[ (cos x - sin x).(cos x - sin x) ]

then i cancelled out (cos x - sin x) as it appeared on the top and bottom line.

But i cant see how one can get 2 / (1 - sin 2 x)

Maybe i need to use log tables? Please provide any information if you can.

Much obliged,
Annie


Answer
From what you have , you can't cancel cosx - sinx from the denominator because it only appears as a factor of one of the terms in the sum in the numerator. However , if you multiply everything out in the numerator, you get

-cosxsinx + cos^2x + sin^2x - sinxcosx + cosxsinx + cos^2x + sin^2x + sinxcosx = 2   (using sin^2x + cos^2x = 1)

In the denominator , if you multiply everything out, you get

cos^2x - cosxsinx - sinxcosx + sin^2x =

1 - 2 sinxcosx =

1 - sin(2x)


So the derivative is in fact

2/(1 - sin2x)