Question

How do you prove `1- [(cos^(2)x)/(1+sinx)]= sinx`?

Answer 1

By the Pythagorean Theorem
`cos^2(x) + sin^2(x) = 1`
or
`cos^2(x) = 1-sin^2(x)`

So
`1-[(cos^2(x))/(1+sin(x))]`

`= 1- [(1-sin^2(x))/(1+sin(x))]`

`=1 - [((1-sin(x))*(1+sin(x)))/(1+sin(x))]`

`= 1- [1-sin(x)]`

`= sin(x)`