How do you simplify $(1 - cos^2 theta)/(1 + sin theta) = sin theta$ ?

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Answer 1 · 2015-11-27T12:12:52

The solutions for this equation are $theta = n pi$ , for $n in ZZ$ .

First, it's a good idea to use the following identity:

$cos^2 theta + sin ^2 theta = 1 color(white)(xx)<=> color(white)(xx) 1 - cos^2 theta = sin^2 theta$

Thus, your equation can bei simplified like follows:

$(sin^2 theta) / (1 + sin theta) = sin theta$

... multiply both sides with the denominator...

$sin^2 theta = (1 + sin theta) sin theta$

$<=> sin^2 theta = sin theta + sin ^2 theta$

$<=> sin theta = 0$

If you graph the $sin$ function, you will see that it intercepts the $x$ axis for ..., $-2 pi$ , $- pi$ , $0$ , $pi$ , $2 pi$ , $3 pi$ , ...

graph{sin x [-10, 10, -2, 2]}

So, your equation is not an identity (and thus it can't be proven as such), but it does have solutions, namely

$theta in n pi$ for $n in ZZ$

How do you simplify (1 - cos^2 theta)/(1 + sin theta) = sin theta(1 - cos^2 theta)/(1 + sin theta) = sin theta?