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I need your help...

apparently this is an relation sin(2x) = tan(x) / (1 + tan^2(x))

that we need to prove that it's not an identity

now if we plug (pi/2) the right side would be = 0 and the other is undefined...

rendering this relation to not be an identity...

now we're asked to fix it...

and apparently the new right equation is sin(2x)= 2 tan(x) / 1 + tan^2(x)

teacher even proved it right through Ls Rs analysis..(which made sense to me when I did it myself)

however when we plug (pi/2) in the fixed one..we still don't get the same result from both sides...instead we get right side=0 and left is undefined again...and as you know

0 doesn't equal undefined

this usually makes a relation not an identity

so in this case, the fix is not an identity either, until restrictions are added to it anyways!

who's right?

help appreciated guys

apparently this is an relation sin(2x) = tan(x) / (1 + tan^2(x))

that we need to prove that it's not an identity

now if we plug (pi/2) the right side would be = 0 and the other is undefined...

rendering this relation to not be an identity...

now we're asked to fix it...

and apparently the new right equation is sin(2x)= 2 tan(x) / 1 + tan^2(x)

teacher even proved it right through Ls Rs analysis..(which made sense to me when I did it myself)

however when we plug (pi/2) in the fixed one..we still don't get the same result from both sides...instead we get right side=0 and left is undefined again...and as you know

0 doesn't equal undefined

this usually makes a relation not an identity

so in this case, the fix is not an identity either, until restrictions are added to it anyways!

who's right?

help appreciated guys

7
replies

yea the first one makes sense....L.S= 2cosxsinx.....RS= 2(sinx/cosx)/ (1+(sin^2x/cos^2x) solve and it equals 2sinxcosx/(sin^2x+cos^2x) which equals 2sinxcosx......the second one doesnt make sense thats the same as saying sin2x=tan2x.

@pj2121 wrote

yea the first one makes sense....L.S= 2cosxsinx.....RS= 2(sinx/cosx)/ (1+(sin^2x/cos^2x) solve and it equals 2sinxcosx/(sin^2x+cos^2x) which equals 2sinxcosx......the second one doesnt make sense thats the same as saying sin2x=tan2x.

There was a typo

care to check it again now?

and can u please tell me if the 2nd one ..would still be an identity after plugging (pi/2)?

as I know an identity must be true for all x values unless there are restrictions which there weren't? am i right?

no what i showed works..just change tan too sin over cos and tan^2 too sin^2 over cos^2 and solve. I dont know how to show it any better.

the problem with plugging in pi/2 is that tan cant equal pi/2 its a restriction. type in tan(pi/2) and you get undefined. As you know if you draw a tan graph tan (pi/2) is a vertical asymptote. Therefore when you plug this value into you calc it reads undefined.

the problem with plugging in pi/2 is that tan cant equal pi/2 its a restriction. type in tan(pi/2) and you get undefined. As you know if you draw a tan graph tan (pi/2) is a vertical asymptote. Therefore when you plug this value into you calc it reads undefined.

@pj2121 wrote

no what i showed works..just change tan too sin over cos and tan^2 too sin^2 over cos^2 and solve. I dont know how to show it any better.

the problem with plugging in pi/2 is that tan cant equal pi/2 its a restriction. type in tan(pi/2) and you get undefined. As you know if you draw a tan graph tan (pi/2) is a vertical asymptote. Therefore when you plug this value into you calc it reads undefined.

True, but

since the right side is undefined

and the left side is 0

at x = (pi/2)

that means this is still not an identity after fixing it, right?

because there are values like (pi/2) and (2pi/3) and such that make both sides not equal to each other...whereas in the definition of an identity, it's required to be true for all values of x unless it came with restrictions to x which this one didn't.

A fix to the first wrong relation should have been:

sin(2x) = 2 tan(x) / (1 + tan^2(x))

in addition to a restriction (x doesn't equal npi + pi/2 where n is an integer)

I hope you understood me now..my only problem is I'm not sure of my logic and I need someone to prove me right or wrong

sorry i misread earlier (oops) read my first response above and it gives you the answer.

But you cant plug in restrictions to prove indentities. Angle x cant equal pi/2 or any undefined angle. But rather then using values I suggest you look at the identity without using values. ie change tan to sin over cos etc. It simplifies the problem. Your teacher is correct she just left out that there are restrictions probably because she doesnt want you to look at the question using values.

But you cant plug in restrictions to prove indentities. Angle x cant equal pi/2 or any undefined angle. But rather then using values I suggest you look at the identity without using values. ie change tan to sin over cos etc. It simplifies the problem. Your teacher is correct she just left out that there are restrictions probably because she doesnt want you to look at the question using values.

@pj2121 wrote

sorry i misread earlier (oops) read my first response above and it gives you the answer.

But you cant plug in restrictions to prove indentities. Angle x cant equal pi/2 or any undefined angle. But rather then using values I suggest you look at the identity without using values. ie change tan to sin over cos etc. It simplifies the problem. Your teacher is correct she just left out that there are restrictions probably because she doesnt want you to look at the question using values.

I plugged in restrictions to prove it's a non-identity which I wouldn't have if there was any restriction on x values mentioned as a part of the relation!!

could a teacher just not mention restrictions that are vital for an equation's validity? just for the point to divert us away from using values to prove it wrong?..

it's still wrong

I know I'm starting to repeat myself..but thanks on all your input, your help was great!