Question

Prove the identity
(csc x - cot x)(sec x + 1) = tan x
And please show me how you did it

Answer 1

\[\csc(x) = \frac{ 1 }{ sin(x) }\]
\[\cot(x) = \frac{ 1 }{ \tan(x) }=\frac{ \cos(x) }{ sin(x) }\]
\[\cos^2(x)=1-\sin^2(x)\]

Answer 2

\[(\csc(x)-\cot(x))=\left( \frac{ 1 }{ \sin(x)}-\frac{ \cos(x) }{ \sin(x) } \right) = \left( \frac{ 1-\cos(x) }{ \sin(x) } \right)\]
\[(\sec(x)+1)=\left( \frac{ 1 }{ \cos(x) } +\frac{ \cos(x) }{ \cos(x) }\right)=\left( \frac{ 1+\cos(x) }{ \cos(x) } \right)\]

Answer 3

\[\left( \frac{ 1-\cos(x) }{ \sin(x) } \right)\left( \frac{ 1+\cos(x) }{ \cos(x) } \right)=\frac{ 1-\cos^2(x) }{ \sin(x)\cos(x) }=\frac{ 1-(1-\sin^2(x) }{ \sin(x)\cos(x) }\]

Answer 4

\[=\frac{ \sin^2(x) }{ \sin(x)\cos(x) }=\frac{ \sin(x) }{ \cos(x) }=\tan(x)\]

Answer 5

Wow...thank you sooo much Lessis! I really understand all that now..You're great :)

Answer 6

I have one more, do you think you can help me with that as well? If you don't mind.

Answer 7

Sure.

Answer 8

And thank you for going step by step

Answer 9

Prove that this is not an identity
sin x + cos x = 1

Answer 10

Do you know how to prove something by contradiction?

Answer 11

No, I don't. This whole proof thing is all very confusing for me.

Answer 12

A proof by contradiction, is when you demonstrate that something is true by assuming it's false and getting a contradiction. As an example, I'll demonstrate that all blue shirt wearing guys, have a blue shirt. Assuming that statement is false, then some blue shirt wearing guys don't have a blue shirt. But if they don't have a blue shirt, then they can't be wearing a blue shirt, which is a contradiction. Then every blue shirt wearing guy MUST have a blue shirt.

Answer 13

Hmm..I see what you're saying.

Answer 14

To prove that sin(x) + cos(x) = 1 isn't an identity, suppose that it actually IS an identity. If that equation is an identity, then for every "x" value you give, sin(x)+cos(x)=1. But for x=pi/4 or 45 degrees, sin(x)+cos(x) equals square root of two. Which is not equal to one, and therefore you have a contradiction!
That means that sin(x)+cos(x)=1 can't be an identity.

Answer 15

Oh, I see! So you basically have to plug in and prove that it isn't an identity

Answer 16

Yes. Because if it were an identity, it would be true for every value of x you give. But you can find an infinite number of values where that isn't true.

Answer 17

Makes sense. Thank you soo much.

Answer 18

You're welcome. ;D