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17. Verify Sin X-Sin X = COS I-coS ris an identity_...

Question

17. Verify Sin X-Sin X = COS I-coS ris an identity_

17. Verify Sin X-Sin X = COS I-coS ris an identity_



Answers

Verify the identity: $$\frac{1}{\sin x \cos x}-\frac{\cos x}{\sin x}=\tan x$$

All right for this exercise, we are going to verify the identity co sign of X times the quantity. And I'm gonna use different bracket shapes here just for clarity. Tan X minus Seeking it negative. X equals sine X minus one. Okay, so, to begin, I am going to convert this tan and seeking ratio into their sign and co sign expressions. Recall that tan X equals sign of X over co sign of X and seeking of X equals one over co sign of X and recall that co sign is an even function, which means that f of negative X equals eth Becks. So the first thing I'm gonna do is I'm gonna take this negative sign out. So this whole expression becomes tan X minus seeking of positive X. And now I'm going to rewrite them in terms of signing co sign. So this is sign of X over co sign of X minus one over co sign of X. And let's bring that co sign turned down again. So because these have the same denominator, we can combine them and we get co sign of X Times Sign of X minus one over co sign of X. Now notice that these co sign terms cancel out and we're left with sign of X minus one, which is what we wanted to verify.

Given an equation, we have very pipe. Whether this equation is an identity or no, we start by considering left inside that is sine X, divided by one minus core sex. First we multiply and divide the sick. Risen by one plus course. X We get this now one bless course X multiplied by one minus scored sex Can Britain as one minus courses Square X So in the numerator we get signed next, multiplied by one plus core sex and ended an interview. Get one minus courses. Choir X one minus courses. Choir X is equal toe sign. Esquire X bigots. I next multiplied by one plus course. X divided by sine Esquire X Sign X's Cancel out one power off each other. We get one plus course X divided by sine X that vote over right hand side. We see that left in sight is equal to the right hand side. This verifies the identity

If you proved under course under minus X minus This I'm the minus X were coaching the course. I'm the X plus. Sign the X where we calm down, funder even identity here cause I'm the minus explain Recorded. Of course I off x because even something from the side off the Manus X because size Ah, what function Therefore we're gonna Manus on the X is in this even out identity we will have now the left hand side we go to for this one, we get a cogent ghosts. I'm the x minus for this. When you get a coach either minus on the X and then we'll get because, you know, goes I I'm the X minus Manners becomes the blessed now sigh off the X and doesn't exactly ico Judah right inside here. Therefore, we prove this identity now

We proved me identity. One minus cost squad X over one person X equals sine X, and I'll be working the left hand side on. My first step is going to be to multiply the numerator and denominator here by one minus sign picks, so I'm going to get one minus. We'll square x times one Mine Sonics divided by one plus I Next times one minor Sonics This is perform a plus B times a minus. B soil gets a square, one minor speed squared Sign Squid X Most Plan B skipper and I'm going to the identity of sine squared X plus close squares. X equals one. So if this is so Ben one of minus signs squared X equals the square ex so I can replace my denominator with cause Square X now gets one minus squared X one minus Sign X, divided by a squared X. Now the's going to cancel out your squad X and I will be left with just one minus one minor son X, which is going to just give me sonics, which is the same as for inside. So we have proven that our identity


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