Prove the Indentity

Question

prove secx+1/tanx=sinx/1-cosx

Answer 1

sec(x)+1=1/cos(x)+1=(1+cos(x))/cos(x);

tan(x)=sin(x)/cos(x) so (sec(x)+1)/tan(x)=

(1+cos(x))/(cos(x)(sin(x)/cos(x))=(1+cos(x))/sin(x).

The given identity is upside down! To prove it, let x=30°; tan(30)=1/√3, cos(30)=√3/2, sin(30)=½.

(sec(x)+1)/tan(x)=(2/√3+1)/(1/√3)=√3(2/√3+1)=2+√3.

(1+cos(x))/sin(x)=(1+√3/2)/½=2+√3. This confirms (sec(x)+1)/tan(x)=(1+cos(x))/sin(x).

sin(x)/(1+cos(x))=½/(1+√3/2)=1/(2+√3)=2-√3, confirming:

(sec(x)+1)/tan(x)≠sin(x)/(1+cos(x)).

However:

sin(x)/(1-cos(x))=sin(x)(1+cos(x))/(1-cos²(x))=

sin(x)(1+cos(x))/sin²(x)=(1+cos(x))/sin(x).

So (sec(x)+1)/tan(x)=sin(x)/(1-cos(x))=(1+cos(x))/sin(x). Maybe this is what you meant.