how do you simplify 1/(1-cosx)-1/sin2x?

Question

How do you simplify:

1/(1-cosx) - 1/sin²x?

Answer 1

--> 1/(1 - cosx) - 1/sin²x

--> 1/(1 - cosx) - 1/(1 - cos²x) [ because sin²x + cos²x = 1]

--> 1/(1 - cosx) - 1/ {(1 - cosx) (1 + cosx)} [use formula a² - b² = (a+b)(a-b) for (1-cos²x)]

--> Take the LCM

--> you get (1+cosx) - 1 as your numerator and (1+cosx)(1-cosx) as your denominator

--> 1 and -1 cancel in the numerator

--> gives you, cosx/{(1+cosx)(1-cosx)}

--> cosx/(1-cos²x) [ going backwards]

--> cosx/sin²x [ going backwards]

--> (cosx/sinx) (1/sinx) [ splitting sin²x into (sinx)(sinx)]

--> (tanx)(cscx)

Hope that makes sense.