Question

How do you verify `(1 - cosx) / (1 + cosx) = (cotx - cscx)²`?

Answer 1

Have a look:

Explanation:

Consider that:
`cot(x)=cos(x)/sin(x)`
and:
`csc(x)=1/sin(x)`
so in your case:
`(1-cos(x))/(1+cos(x))=(cos(x)/sin(x)-1/sin(x))^2`
`(1-cos(x))/(1+cos(x))=((cos(x)-1)/sin(x))^2`
`(1-cos(x))/(1+cos(x))=(cos(x)-1)^2/sin^2(x)`
but simplyfying and from:
`1=sin^2(x)+cos^2(x)`
`sin^2(x)=1-cos^2(x)`
`cancel((1-cos(x)))/(1+cos(x))=(cos(x)-1)^cancel(2)/(1-cos^2(x))`
also:
`(1-cos^2(x))=(1+cos(x))(1-cos(x))`
`1/(1+cos(x))=(cos(x)-1)/((1+cos(x))(1-cos(x)))`
and finally:
`1/cancel((1+cos(x)))=cancel((cos(x)-1))/(cancel((1+cos(x)))cancel((1-cos(x))))`
or:
`1=1`