Question

PLEASE HELP!!

Prove and match the ones that are the same

1. csc x − sin x
2. 2 sin(4x) cos(4x)
3. sin x −1/sin x +1

a. −cos^2 x/(sin x+1)^2
b. cosxcotx
c. sin(8x)

Answer 1

`2.=c.\\\\2\sin4x\cos4x=2\sin(2\cdot4x)=2\sin8x\\\\Used:\\\sin2\alpha=2\sin\alpha\cos\alpha`


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


`3.=a.\\\\(\sin x-1)/(\sin x+1)=(\sin x-1)/(\sin x+1)\cdot(\sin x+1)/(\sin x+1)=(\sin^2x-1^2)/((\sin x+1)^2)=(\sin^2x-1)/((\sin x+1)^2)\\\\=(-(1-\sin^2x))/((\sin x+1)^2)=(-\cos^2x)/((\sin x+1)^2)\\\\Used:\\(a-b)(a+b)=a^2-b^2\\\\\sin^2x+\cos^2x=1\to \cos^2x=1-\sin^2x`