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#