Proving the Relationship between Inverse Sine and Cosine Functions

[srini]

To show that
cos-1(-x)-cos-1(x)=2sin-1(x)

I tried
take x= sina
taking cos of the whole equation
cos(cos-1(-x))-cos(cos-1(x))=2cos(sin-1(x))
now we have to prove : -x-x=2cos(sin-1(x))
LHS: -2x=-2sina=2cos(a+pi/2)
RHS: 2cosa

Iam not sure how to proceed further..can anyone help me with this..

 

[marcusl]

To show that
cos-1(-x)-cos-1(x)=2sin-1(x)

I tried
take x= sina
taking cos of the whole equation
cos(cos-1(-x))-cos(cos-1(x))=2cos(sin-1(x))

This equation is incorrect. You need to expand cos(arcos(-x) - arcos(x)) properly,
You will then need to use relations like cos(arsin(y))=sqrt(1-y^2)

now we have to prove : -x-x=2cos(sin-1(x))
LHS: -2x=-2sina=2cos(a+pi/2)
RHS: 2cosa

Iam not sure how to proceed further..can anyone help me with this..

 

[Dick]

You could also proceed more concretely. Get out (or make) a graph of cos^(-1) and sin^(-1) (let's call them acos and asin). If cos(theta)=x then cos(pi-theta)=(-x). So acos(-x)-acos(x)=pi-2*theta. Now if cos(theta)=x then sin(pi/2-theta)=x. So asin(x)=?.