If y=arcsine (cos X) prove that tan(X/2)=Sec y - Tany

Question

I have managed to reach so far

y=90 - x

Sec(90-x)-tan(90-x)=csc(X)-cot(X)=

1-cos(X)/sin(X)

Answer 1

We have

y=arcsine (cos x)

From here it follows that

sin(y)=cos(x)     (*)

 

Next, we have to show that tan(X/2)=Sec y - Tan y

sec(y) - tan(y)=1/cos(y)-sin(y)/cos(y)=

=(1-sin(y))/cos(y)= 

 

=|due to (*) sin(y)=cos(x) and cos(y)=sin(x)|=

 

=(1-cos(x))/sin(x)=Tan(x/2)

Q.E.D