Question

Verify the identity.

cotx minus pi divided by two. = -tan x

Answer 1

HI!!

Answer 2

if you can use the cofunction identity this is real easy

Answer 3

since cotangnent is even, you know
\[\cot(x-\frac{\pi}{2})=-\cos(\frac{\pi}{2}-x)\]

Answer 4

damn i meant "odd"

Answer 5

in any case , as cotangent is odd, you have
\[\cot(x-\frac{\pi}{2})=-\cos(\frac{\pi}{2}-x)\] and the cofunction identity tells gives
\[\cos(\frac{\pi}{2}-x)=\tan(x)\] and you are done

Answer 6

yea, its tan x right?

Answer 7

damn i am making a lot of typos

Answer 8

let me try again in one line
\[\cot(x-\frac{\pi}{2})=-\cot(\frac{\pi}{2}-x)=-\tan(x)\]

Answer 9

yea, -tan x right?

Answer 10

the first equal sign because cotangent is odd
the second equal sign in the cofunction identity

Answer 11

is this clear? it is really only two steps, but maybe they are not obvious

Answer 12

the identity is -tan x or am I lost.

Answer 13

Is it cos(pi/2-x)
?

Answer 14

i thought that might be the case
lets go slow and take them one at a time
the cofunction identities say \[\cos(\frac{\pi}{2}-x)=\sin(x)\\
\csc(\frac{\pi}{2}-x)=\sec(x)\\
\cot(\frac{\pi}{2}-x)=\tan(x)\]

Answer 15

cot(pi/2-x)

Answer 16

since \[\cot(\frac{\pi}{2}-x)=\tan(x)\] that is the same as
\[-\cot(\frac{\pi}{2}-x)=-\tan(x)\] just change the sign of both sides

Answer 17

is that part clear?

Answer 18

Oh ok, yes it is.
Thnx!! :)

Answer 19

ok and how about the first part
\[\cot(x-\frac{\pi}{2})=-\cot(\frac{\pi}{2}-x)\]?

Answer 20

yea, that's the part that kinda got me.

Answer 21

ok lets do that slow too

Answer 22

Ok :)

Answer 23

cotangent in an "odd" function, which means
\[\cot(-x)=-\cot(x)\] in general "odd" means
\[f(-x)=-f(x)\]

Answer 24

sine and tangent are also "odd" so
\[\sin(-x)=-\sin(x)\] etc

Answer 25

now
\[-(x-\frac{\pi}{2})=\frac{\pi}{2}-x\] right ?

Answer 26

Yes...

Answer 27

that means, since cotangent is odd, that
\[\cot(x-\frac{\pi}{2})=\cot(\color{red}-(\frac{\pi}{2}-x))=\color{red}-\cos(\frac{\pi}{2}-x)\]

Answer 28

notice the minus sign comes out of the function

Answer 29

oh yea, this makes sense....

Answer 30

it is a bit abstract, but not too hard
just realizing that \(a-b=-(b-a)\) and therefore \(\cot(a-b)=-\cos(b-a)\)

Answer 31

oh, wow it's so much clearer now. Thank you @misty1212 for your help.

Answer 32

\[\huge \color\magenta\heartsuit\]

Answer 33

<3

Answer 34

Lol

Answer 35

thanks