what is cos(x)cot(x)=cosec(x)-sin(x)

Question

how do i prove it, i have tried everything and tried all the identities it only comes as cos + cot not cos multiplied by cot. i am definately doing something wrong but i dont know what. but i do know the i am close to the answer however.

Answer 1

If (a,b) on terminal ray of angle x,

then (rcosx,rsinx) := (a,b) where

r = √(a^2+b^2).

 

Then

 

sinx = b/r

cosx = a/r

tanx := sinx/cosx = b/a

cotx := 1/tanx = a/b

secx := 1/cosx = r/a

cscx := 1/sinx = r/b

 

cos(x)cot(x) =? csc(x) - sin(x)

(a/r)(a/b) =? (r/b) - (b/r)

a^2/(br) =? (r^2-b^2)/(br)

a^2/(br) =? (a^2+b^2-b^2)/(br)

a^2/(br) = a^2/(br) √

Answer 2

Hi Sally - use the cos²(x)=1-sin²(x) identity on the left after setting cot(x)=cos(x)/sin(x) then multiplying by cos(x)

So transform the left side to result in the right side.  That is how the subtraction shows up.