d/dx (cosx+sinx/cosx-sinx)
I'm having a bit of trouble with trying to prove LHS = RHS (or the other way around for that matter)
I have simplified the LHS to be 2/1-2cosxsinx but how do i get the RHS?
d/dx (cosx+sinx/cosx-sinx)
I'm having a bit of trouble with trying to prove LHS = RHS (or the other way around for that matter)
I have simplified the LHS to be 2/1-2cosxsinx but how do i get the RHS?
Dear
Yep. I must be correct, since using the above expression I managed to show that the RHS=LHS.
First consider,
$\displaystyle \frac{\cos{x} + \sin{x}}{\cos{x} - \sin{x}}=\frac{sin\left(\frac{\pi}{2}-x\right)+sinx}{sin\left(\frac{\pi}{2}-x\right)-sinx}=\frac{2sin\frac{\pi}{4}cos\left(\frac{\pi}{4 }-x\right)}{2cos\frac{\pi}{4}sin\left(\frac{\pi}{4}-x\right)}=\frac{cos\left(\frac{\pi}{4}-x\right)}{sin\left(\frac{\pi}{4}-x\right)}$
$\displaystyle =\tan\left(\frac{\pi}{2}-\frac{\pi}{4}+x\right)$
I hope you could continue from here!!
Could i ask, how did you get:
$\displaystyle \frac{2sin\frac{\pi}{4}cos\left(\frac{\pi}{4}-x\right)}{2cos\frac{\pi}{4}sin\left(\frac{\pi}{4}-x\right)}$
and also i think you meant:
$\displaystyle =\cot\left(\frac{\pi}{2}-\frac{\pi}{4}+x\right)$
instead of tan? please correct me if i'm mistaken.
Dear ipokeyou,
1) Do you know the sum-to-product rule?? If not better refer: Sum-to-Product formulas (Proof of these formulas could be found in many basic trignometric books.)
2) $\displaystyle \frac{cos\left(\frac{\pi}{4}-x\right)}{sin\left(\frac{\pi}{4}-x\right)}=\cot\left(\frac{\pi}{4}-x\right)=\tan\left(\frac{\pi}{2}-\frac{\pi}{4}+x\right)$
Does this solve your problems?? If not please don't hesitate to reply.
ah yes, its just i havent learnt the sum-to-product rules yet but i get it. would there be a way to simplify it without it? just wondering.
and
i still dont get how you get from $\displaystyle \cot\left(\frac{\pi}{4}-x\right)=\tan\left(\frac{\pi}{2}-\frac{\pi}{4}+x\right)$
sorry for these silly questions
See Complementary Angle Identities: Proofs of trigonometric identities - Wikipedia, the free encyclopedia