Question

Prove the identity: (1-sin(x))/(1+sin(x)) = tan(x) + 1/cos(x)

Answer 1

\[\frac{1-\sin(x)}{1+\sin(x)} \cdot \frac{1-\sin(x)}{1-\sin(x)}=\frac{1-2\sin(x)+\sin^2(x)}{1-\sin^2(x)}=\frac{1-2 \sin(x)+\sin^2(x)}{\cos^2(x)}\]
\[\frac{1}{\cos^2(x)}-2 \frac{\sin(x)}{\cos^2(x)} +\tan^2(x)=\sec^2(x)-2 \frac{\sin(x)}{\cos(x)} \cdot \frac{1}{\cos(x)}+\tan^2(x)\]
\[=\sec^2(x)-2 \tan(x) \sec(x)+\tan^2(x)\]

Answer 2

i don't see how it can be written as tan(x) + 1/cos(x)

Answer 3

So are you saying it's not an identity? :|

Answer 4

wait i'm looking
it is tan(x)+1/cos(x) right?

Answer 5

tan(x) + (1/cos(x))

Answer 6

\[=(\sec(x)-\tan(x))^2\]

Answer 7

\[=(1)^2=1\]

Answer 8

sumpin wrong here

Answer 9

I've checked wolframalpha and it is indeed an identity that CAN be proven; I'm really stumped though.

Answer 10

did i make a mistake?

Answer 11

here is my proof
let
\[x=\frac{\pi}{4}\] then
\[\frac{1-\frac{\sqrt{2}}{2}}{1+\frac{\sqrt{2}}{2}}\]is the left hand side, whereas the right hand side is
\[1+\sqrt{2}\]

Answer 12

:|

Answer 13

and this would be a miracle, since the left hand side is smaller than one, and the right hand side is larger

Answer 14

so you ought to be stumped, because i cannot be correct.

Answer 15

sigh..Alright haha

Answer 16

Thanks anyways

Answer 17

hey don't believe everything you read in a text book...

Answer 18

I want to believe this is possible because Wolframlpham says it's a valid identity; but I guess we cannot solve it >_<

Answer 19

actually wolfram does not say it is an identity. it solved the equation for me here
http://www.wolframalpha.com/input/?i=%281-sin%28x%29%29%2F%281%2Bsin%28x%29%29%3Dtan%28x%29+%2B+1%2Fcos%28x%29%2C+

Answer 20

put "verify" infront of the entire identity

Answer 21

hold the phone. it is not an identity for sure. i proved it, but wolfram solves the equation for x, which means it is certainly not true for all x

Answer 22

when i put "verify" it just timed out

Answer 23

Hmm..So I should conclude that this is not an identity after all?

Answer 24

hey why is that one theta bolder type than the other thetas

Answer 25

yes we concluded that way up there

Answer 26

identity means "true for any value of x" like
\[2x+x=3x\] is an identity. this one is not

Answer 27

What if it was a theata instead of "x"

Answer 28

\[2 \theta+\theta=3 \theta? \]
this is still an identity

Answer 29

2p+p=3p is still an identity

Answer 30

Alright, thanks for the evidence that indeed this is not an identity :)