Question

HELP NEEDED!!! WILL FAN AND MEDAL!!!! I need help understanding how to prove identities.

Answer 1

I have a specific example. Please help...

Answer 2

what is it?

Answer 3

\[\tan \frac{ \cos \theta }{ 1+ \sin \theta }=\frac{ 1 }{ \cos \theta }\]

Answer 4

It was actually supposed to be tangent theta times cosine theta over one plus sine theta

Answer 5

I have all of the identities written out so I can easily refer to them, but the problem I am having is learning how to combine the like terms.

Answer 6

\[\frac{ \sin \theta }{ \cos \theta }+\frac{ \cos \theta }{ 1+ \sin \theta }\] this is what I need help combining because I have already converted Tan

Answer 7

what the heck is the tan doinog there ?

Answer 8

The tan was there because our teacher was trying to get us to recreate all of the identities to where they equal one another

Answer 9

no don't convert to tangent
in fact, if it was tangent, you would convert to \(\frac{\sin(x)}{\cos(x)}\)

Answer 10

lets do a tiny bit of algebra first

Answer 11

That's what I did and that was my farthest point in this.

Answer 12

ok i got that wrong, hold the phone

Answer 13

\[\frac{b}{a}+\frac{a}{1+b}\] that's better
can you add these?

Answer 14

I have no idea how to... or at least I can't remember. I know with multiplying you foil it, if you divide you use the kcf format :/ but adding and subtracting makes no sense to me because all I know is a common denominator is required

Answer 15

lol that is always the problem, the algebra

Answer 16

here is the one true way to add fractions \[\huge \frac{A}{B}+\frac{C}{D}=\frac{AD+BC}{BD}\]

Answer 17

o.o"

Answer 18

in your case you will have \[\frac{b(1+b)+a^2}{a(1+b)}\]

Answer 19

I am getting more lost by the minute...

Answer 20

that is the way to add fractions if they are numbers\[\frac{2}{7}+\frac{3}{5}=\frac{2\times 5+7\times 3}{7\times 5}\]

Answer 21

and it is also the way to add fractions if they have variables
it is the only way to do this, by using algebra to add

Answer 22

ohhhh ok, I get it, sort of.

Answer 23

forget that "least common denominator" nonsense you were taught
that is the way to add, you cannot avoid it

Answer 24

so ready to start again?

Answer 25

Yeah I suppose

Answer 26

using the one true way to add \[\frac{b}{a}+\frac{a}{1+b}\] you get \[\frac{b(1+b)+a^2}{a(1+b)}\]

Answer 27

now some more algebra, but just a little \[\frac{b(1+b)+a^2}{a(1+b)}=\frac{b+b^2+a^2}{a(1+b)}\]

Answer 28

that is just the distributive law in the numerator
now that we are done with algebra, we can go back to sines and cosines

Answer 29

sooooooo then \[\frac{ \cos \theta }{ \sin \theta } + \frac{ \cos \theta }{ 1+\sin \theta }\]
should become....

Answer 30

\[\frac{\sin(x)+\sin^2(x)+\cos^2(x)}{\cos(x)(1+\sin(x))}\]

Answer 31

Ok. But now comes an even trickier part.... I have to find a way to reduce that

Answer 32

hold a sec
the initial question was \[\frac{ \sin \theta }{ \cos \theta }+\frac{ \cos \theta }{ 1+ \sin \theta }\] right?

Answer 33

Yes

Answer 34

so after the algebra we get to \[\frac{\sin(x)+\sin^2(x)+\cos^2(x)}{\cos(x)(1+\sin(x))}\] don't look to reduce yet

Answer 35

the usual miracle occurs, \[\sin^2(x)+\cos^2(x)=1\] the mother of all trig identities

Answer 36

correct

Answer 37

The reasoning behind adding and subtracting the fractions is exactly what was said above, but the step being left out from above that might make it clear is that you just multiply both fractions by a number that will make both denominators the same or "common". In the above example....\[\frac{ b }{ a }+\frac{ a }{ 1+b } = (\frac{ 1+b }{ 1+b })(\frac{ b }{ a })+(\frac{ a }{ a })(\frac{ a }{ 1+b })\] and that is where the other equation @satellite73 came up with came from. Not to inturrupt

Answer 38

\[\frac{\sin(x)+\overbrace{\sin^2(x)+\cos^2(x)}^{\text{ this is 1}}}{\cos(x)(1+\sin(x))}\]

Answer 39

\[\frac{\sin(x)+1}{\cos(x)(1+\sin(x))}\] NOW you can cancel the common factor of \[1+\sin(x)\] top and bottom

Answer 40

Ok, ok, I see it now. so the Sins actually match and leave you with \[\frac{ 1 }{ \cos \theta }\]

Answer 41

yes
if you want to impress your teacher, write it as \[\sec(\theta)\]

Answer 42

i hope you also see that 98% of this is algebra

Answer 43

which, if your algebra is not what it needs to be, is going to be a problem, so bone up on it
also use the gimmick of replacing sine and cosine by letters to make the algebra easier
if we had to do this writing sine and cosine each time it would have been a pain