Question

Use a graph in a [-2π, 2π, π/2] by [-3, 3, 1] viewing rectangle to complete the identity

Answer 1

We have the expression:

$\frac{1-2\cos 2x}{2\sin x-1}$

Let's work first with the numerator. We have that we can write cos2x like this:

$\cos 2x=1-\sin ^2x$

doing this substitution we get the following:

$\begin{gathered} 1-2\cos 2x=1-2\cdot(1-\sin ^2x)=1-2+4\sin ^2x=4\sin ^2x-1 \\ =(2\sin x+1)(2\sin x-1) \end{gathered}$

Now that we have these two factors, we can use them on the original identity to get:

$\frac{1-2\cos2x}{2\sin x-1}=\frac{(2\sin x+1)(2\sin x-1)}{2\sin x-1}=2\sin x+1$

therefore, the resulting identity is 2sinx+1