Question

Prove these two

1.) Sin x / 1-cos x + sin x / 1+ cos x = 2 csc x

2.) - tan2x + sec2x = 1

Answer 1

For the first one, we have

`\frac1{1-\cos x}+\frac1{1+\cos x}=(1+\cos x)/((1-\cos x)(1+\cos x))+(1-\cos x)/((1-\cos x)(1+\cos x))`

`=(1+\cos x+1-\cos x)/(1-\cos^2x)`

`=\frac2{\sin^2x}`

Multiply this by
`\sin x` and you end up with

`=(2\sin x)/(\sin^2x)=\frac2{\sin x}=2\csc x`

For the second one,

`-\tan^2x+\sec^2x=-(\sin^2x)/(\cos^2x)+\frac1{\cos^2x}=(1-\sin^2x)/(\cos^2x)=(\cos^2x)/(\cos^2x)=1`