SOLUTION: Prove the identity: {{{-Cotx + Sinx/(1-Cosx) =Cscx}}}

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Question 425709: Prove the identity:
-Cotx+%2B+Sinx%2F%281-Cosx%29+=Cscx

Answer by AnlytcPhil(1806) About Me  (Show Source):
You can put this solution on YOUR website!
Prove the identity:
-Cotx+%2B+Sinx%2F%281-Cosx%29+=Cscx


We will work with just the left side:

Use the quotient identity Cot%28theta%29=Cos%28theta%29%2FSin%28theta%29
to rewrite the first term:

-%28Cosx%29%2F%28Sinx%29+%2B+Sinx%2F%281-Cosx%29


Multiply the second fraction by red%28%28%281%2BCosx%29%29%2F%28%281%2BCosx%29%29%29
which just equals 1:


 

Multiply the fractions:

-%28Cosx%29%2F%28Sinx%29+%2B+%28Sinx%281%2BCosx%29%29%2F%281-Cos%5E2x%29

Use the Pythagorean identity Sin%5E2theta%2BCos%5E2theta=1 written as 1-Cos%5E2theta=Sin%5E2theta
to replace the second denominator:

-%28Cosx%29%2F%28Sinx%29+%2B+%28Sinx%281%2BCosx%29%29%2F%28Sin%5E2x%29

Simplify the second term on the bottom:

-%28Cosx%29%2F%28Sinx%29+%2B+%28cross%28Sinx%29%281%2BCosx%29%29%2F%28Sin%5Ecross%282%29x%29

-%28Cosx%29%2F%28Sinx%29+%2B+%281%2BCosx%29%2F%28Sinx%29

The denominators are the same so we can combine the
two fractions by combining the numerators over the
common denominator:

%28-Cosx%2B1%2BCosx%29%2F%28Sinx%29

Simplify the numerator:

%28cross%28-Cosx%29%2B1%2Bcross%28Cosx%29%29%2F%28Sinx%29

1%2F%28Sinx%29

Use the reciprocal identity Csc%28theta%29=1%2FSin%28theta%29

Csc%28x%29

Edwin