Question

20 POINTS!

Verify the identity.

Answer 1

`(\cot x)/(1+\csc x)=(\csc x-1)/(\cot x)\\\\L_s=(\cos x)/(\sin x):\left(1+(1)/(\sin x)\right)=(\cos x)/(\sin x):\left((\sin x)/(\sin x)+(1)/(\sin x)\right)\\\\=(\cos x)/(\sin x):(\sin x+1)/(\sin x)=(\cos x)/(\sin x)\cdot(\sin x)/(\sin x+1)=(\cos x)/(\sin x+1)`


`R_s=\left((1)/(\sin x)-1\right):(\cos x)/(\sin x)=\left((1)/(\sin x)-(\sin x)/(\sin x)\right)\cdot(\sin x)/(\cos x)\\\\=(1-\sin x)/(\sin x)\cdot(\sin x)/(\cos x)=(1-\sin x)/(\cos x)\cdot(1+\sin x)/(1+\sin x)=(1-\sin^2x)/(\cos x(1+\sin x))\\\\=(\cos^2x)/(\cos x(1+\sin x))=(\cos x)/(1+\sin x) \\\\L_s=R_s`


`Used:\\\csc x=(1)/(\sin x)\\\\\cot x=(\cos x)/(\sin x)\\\\\sin^2x+\cos^2x=1\to\cos^2x=1-\sin^2x\\\\(a-b)(a+b)=a^2-b^2`