How do you verify the identity `tanx csc^2x-tanx=cotx` ?

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Hello!

Recall the definitions of tan, csc and cot:

`tan(x)=sin(x)/cos(x),`  `csc(x)=1/sin(x), cot(x)=cos(x)/sin(x).`

Therefore the left part is equal to

`sin(x)/cos(x) * 1/(sin^2(x))-sin(x)/cos(x) =sin(x)/cos(x) (1/(sin^2(x))-1)=`

`=sin(x)/cos(x) * (1-sin^2(x))/(sin^2(x)) =sin(x)/cos(x) * (cos^2(x))/(sin^2(x))=cos(x)/sin(x),`

which is really equal to the right part, Q.E.D.

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Hello!

Recall the definitions of tan, csc and cot:

`tan(x)=sin(x)/cos(x),`  `csc(x)=1/sin(x), cot(x)=cos(x)/sin(x).`

 

Therefore the left part is equal to

`sin(x)/cos(x) * 1/(sin^2(x))-sin(x)/cos(x) =sin(x)/cos(x) (1/(sin^2(x))-1)=`

`=sin(x)/cos(x) * (1-sin^2(x))/(sin^2(x)) =sin(x)/cos(x) * (cos^2(x))/(sin^2(x))=cos(x)/sin(x),`

 

which is really equal to the right part, Q.E.D.

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