prove identity

Question

Sin/ 1-cosx - 1/sin= 1/tanx   prove the identity  

Answer 1

(sinx)/(1-cosx)-(1/sinx)=(1/tanx)

write the left side as a fraction

(sinx)(sinx)/(sinx)(1-cosx)-(1)(1-cosx)/(sinx)(1-cosx)=1/tanx

[sin²x-1+cosx)/(sinx)(1-cosx)=1/tanx

(sin²x-1+cosx)/(sinx-sinxcosx)=1/tanx

sin²x-1=-cos²x

(-cos²x+cosx)/(sinx-sinxcosx)=1/tanx

factor

(cosx)(1-cosx)/(sinx)(1-cosx)=1/tanx

simplify the left side

cosx/sinx=1/tanx

since tanx=sinx/cosx, 1/tanx=cosx/sinx

1/tanx=1/tanx

 

 

Answer 2

Hi Angel,

 

OK   I will write this as I think you intended it

 

sin x/(1-cos x) - 1/sin x = 1/ tan x

 

First I will multiply both sides by (1 - cos x)(sin x)  to get rid of the fractions on the left

 

sin² x - (1 - cos x) = (1 - cos x)(sin x) /tan x

 

sin² x - 1 + cosx = (1 - cos x)(sin x) /tan x      (but sin² x -1 = -cos² x  so

 

-cos² x   + cos x = (1 - cos x)(sin x) / (sin x/cos x)   ( now I will invert the denominator and multiply on the right side

 

-cos² x + cos x =  (1 - cos x)(sin x) * cos x/sin x   ( the sin x in the numerator and denominator cancel out

 

-cos² x + cos x =  (1 - cos x) * cos x

 

-cos² x + cos x = cos x - cos² x

 

if you rearrange the terms these are in fact the same

 

Hope this helps