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

1 Answer
Gió
Nov 1, 2015

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

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