Sure, let's solve the expression step-by-step.
We are given:
[tex]\[ \frac{\sin \theta - 2 \sin^3 \theta}{2 \cos^3 \theta - \cos \theta} \][/tex]
1. Identify the terms in the expression:
The numerator is: \(\sin \theta - 2 \sin^3 \theta\)
The denominator is: \(2 \cos^3 \theta - \cos \theta\)
2. Factorize the expressions, if possible:
Let's look at the numerator first:
[tex]\[
\sin \theta - 2 \sin^3 \theta = \sin \theta (1 - 2 \sin^2 \theta)
\][/tex]
Now, let's see the denominator:
[tex]\[
2 \cos^3 \theta - \cos \theta = \cos \theta (2 \cos^2 \theta - 1)
\][/tex]
3. Rewrite the fraction using the factored terms:
Substituting the factored forms, we get:
[tex]\[
\frac{\sin \theta (1 - 2 \sin^2 \theta)}{\cos \theta (2 \cos^2 \theta - 1)}
\][/tex]
4. Simplify the expression:
We notice that \(2 \cos^2 \theta - 1\) can also be related to the double angle identity for cosine:
[tex]\[
2 \cos^2 \theta - 1 = \cos 2 \theta
\][/tex]
So the denominator becomes:
[tex]\[
\cos \theta \cdot \cos(2\theta)
\][/tex]
And recognize that \(1 - 2 \sin^2 \theta\) is related to the double angle identity for sine:
[tex]\[
1 - 2 \sin^2 \theta = \cos(2\theta)
\][/tex]
Hence, our numerator becomes:
[tex]\[\sin \theta \cdot \cos(2\theta)\][/tex]
5. Final simplified form:
Combining both results, the expression simplifies nicely to:
[tex]\[
\frac{\sin \theta \cos (2 \theta)}{\cos \theta \cos (2 \theta)}
\][/tex]
We see that \(\cos (2 \theta)\) appears in both the numerator and denominator. Thus, it can be cancelled out (assuming \(\cos (2 \theta) \ne 0\)), leaving us with:
[tex]\[
\frac{\sin \theta}{\cos \theta}
\][/tex]
6. Conclude the simplified form:
The simplified form of the original expression is:
[tex]\[
\tan \theta
\][/tex]
