Question

How do you prove cotx/(cscx-sinx)=secx

Answer 1

`(\cot x)/(\csc x-\sin x)=((\cos x)/(\sin x))/((1)/(\sin x)-\sin x)= ((\cos x)/(\sin x))/((1)/(\sin x)-(\sin^2x)/(\sin x))= ((\cos x)/(\sin x))/((1-\sin^2x)/(\sin x))=\\\\\\= ((\cos x)/(\sin x))/((\cos^2x)/(\sin x))=(\cos x\cdot\sin x)/(\sin x\cdot\cos^2x)=(1)/(\cos x)=\boxed{\sec x}`