Ask your own question, for FREE!
Mathematics 83 Online
OpenStudy (gold):

prove: sinx+cosx/tan^2x-1 = cos^2x/sinx-cosx

OpenStudy (gold):

\[\sin ^{4}x - \cos ^{4}x/ \tan ^{4}x -1 = \cos ^2x\]

OpenStudy (shadowfiend):

I'm going to look at the first equation for now. Is the second a different proof that you need to do? There are two things to keep in mind here. First off, remember when trying to prove these that you can work both sides (it will be easier). Secondly, remember that \(\tan x = \frac{\sin x}{\cos x}\), which means that \(\tan^2 x = \frac{\sin^2 x}{\cos^2 x}\). So, once we've determined that, we can first take \(\sin x - \cos x\) from the right and multiply it into the left: \[\begin{align} \frac{\sin x + \cos x}{\tan^2 x - 1} &= \frac{\cos^2 x}{\sin x - \cos x}\\ \frac{(\sin x + \cos x)(\sin x - \cos x)}{\tan^2 x - 1} &= \cos^2 x\\ \frac{\sin^2 x - \cos^2 x}{\tan^2 x - 1} &= \cos^2 x \end{align}\] You can then take \(\tan^2 x - 1\) and multiply it into the right: \[\begin{align} \frac{\sin^2 x - \cos^2 x}{\tan^2 x - 1} &= \cos^2 x\\ \sin^2 x - \cos^2 x &= \cos^2 x(\tan^2 x - 1) \end{align}\] Then we use the fact that \(\tan^2 x = \frac{\sin^2 x}{\cos^2 x}\) and multiply through by \(\cos^2 x\) on the right: \[\begin{align} \sin^2 x - \cos^2 x &= \cos^2 x(\tan^2 x - 1)\\ &=\cos^2 x \tan^2 x - \cos^2 x\\ &= \cos^2 x\frac{\sin^2 x}{\cos^2 x} - \cos^2 x\\ \sin^2 x - \cos^2 x = \sin^2 x - \cos^2 x \end{align}\] When we expand \(\tan^2 x\), we cancel out the \(\cos^2 x\) and we are left with the same expressions on both sides, which means we have proven the identity.

Can't find your answer? Make a FREE account and ask your own questions, OR help others and earn volunteer hours!

Join our real-time social learning platform and learn together with your friends!
Latest Questions
desss: are any of yall good at mathh!!??
1 minute ago 13 Replies 1 Medal
demonwolftwin: who was the 2rd presseddent
2 hours ago 1 Reply 0 Medals
Tbone: I made one last edit I'ma stop making them for a while
42 minutes ago 22 Replies 8 Medals
yomomalikebracobama: e
5 hours ago 6 Replies 1 Medal
yomomalikebracobama: help u tigers
5 hours ago 10 Replies 2 Medals
Williammcgeorge: does anyone want to see some of my drawings
2 hours ago 19 Replies 4 Medals
Joe348: Math 4 Vectors/Vectors Applications
6 hours ago 2 Replies 1 Medal
Joe348: Math :(
6 hours ago 20 Replies 4 Medals
Can't find your answer? Make a FREE account and ask your own questions, OR help others and earn volunteer hours!

Join our real-time social learning platform and learn together with your friends!