math_knight
New member
- Joined
- Jun 13, 2018
- Messages
- 13
so the book is talking about proving an identity as follows:
(sorry I wish I knew how to use LaTex font)
sin^2(-t) - cos^2(-t) / sin(-t)-cos(-t) = cos (t) - sin (t)
using even/odd identities, and looking at only the left side and the numerator of the equation it becomes
Numerator : -sin^2(t) - cos^2(t)
or
[-sin t]^2 - [cos t]^2 (difference of squares)
but the book writes it as
[sin t]^2 - [cos t]^2
Where did the negative sign go?
OK, wait, I thought about it some, but just want to make sure I did it correctly. I guess they factored out (-1) TWICE?
so that
(-sin t - cos t) (-sin t + cost t)
= (-1)(sin t + cost t) (-1)(sin t - cos t)
=(-1)(-1)(sin t + cost t)(-1)(sin t - cos t)
=(1)(sin t + cost t)(sin t - cos t)
=(sin t + cost t)(sin t - cos t)
Is that legal in all 50 states? I guess they are factoring out (1) so the sign has been accounted for. I have seen (-1) get factored out before, just never twice like that, if that's in fact what they did.
***It's still hard to see how they got [sin t]^2 - [cos t]^2 from [-sin t]^2 - [cos t]^2 though??
I would think you would have to wait to factor out (-1) twice only after you factored the difference of squares, which is what I did in the orange equations. How did they factor out (-) sign where they did?
Any help would be really appreciated.
(sorry I wish I knew how to use LaTex font)
sin^2(-t) - cos^2(-t) / sin(-t)-cos(-t) = cos (t) - sin (t)
using even/odd identities, and looking at only the left side and the numerator of the equation it becomes
Numerator : -sin^2(t) - cos^2(t)
or
[-sin t]^2 - [cos t]^2 (difference of squares)
but the book writes it as
[sin t]^2 - [cos t]^2
Where did the negative sign go?
OK, wait, I thought about it some, but just want to make sure I did it correctly. I guess they factored out (-1) TWICE?
so that
(-sin t - cos t) (-sin t + cost t)
= (-1)(sin t + cost t) (-1)(sin t - cos t)
=(-1)(-1)(sin t + cost t)(-1)(sin t - cos t)
=(1)(sin t + cost t)(sin t - cos t)
=(sin t + cost t)(sin t - cos t)
Is that legal in all 50 states? I guess they are factoring out (1) so the sign has been accounted for. I have seen (-1) get factored out before, just never twice like that, if that's in fact what they did.
***It's still hard to see how they got [sin t]^2 - [cos t]^2 from [-sin t]^2 - [cos t]^2 though??
I would think you would have to wait to factor out (-1) twice only after you factored the difference of squares, which is what I did in the orange equations. How did they factor out (-) sign where they did?
Any help would be really appreciated.