Certainly! Let's go through the problem step by step.
We are given the expression:
[tex]\[
\tan \theta \cdot (1 - \tan \theta) \cdot (1 + \tan^2 \theta)
\][/tex]
Let's denote [tex]\(\tan \theta\)[/tex] by [tex]\( t \)[/tex] for simplicity. So the expression becomes:
[tex]\[
t \cdot (1 - t) \cdot (1 + t^2)
\][/tex]
Now, we'll simplify the expression step by step.
1. First, consider [tex]\( t \cdot (1 - t) \)[/tex]:
[tex]\[
t \cdot (1 - t) = t - t^2
\][/tex]
2. Next, multiply the result by [tex]\( (1 + t^2) \)[/tex]:
[tex]\[
(t - t^2) \cdot (1 + t^2)
\][/tex]
Let's distribute each term in [tex]\( (t - t^2) \)[/tex] through the [tex]\( (1 + t^2) \)[/tex]:
3. Distribute the [tex]\( t \)[/tex] term:
[tex]\[
t \cdot (1 + t^2) = t \cdot 1 + t \cdot t^2 = t + t^3
\][/tex]
4. Distribute the [tex]\( -t^2 \)[/tex] term:
[tex]\[
-t^2 \cdot (1 + t^2) = -t^2 \cdot 1 + -t^2 \cdot t^2 = -t^2 - t^4
\][/tex]
5. Combine all the distributed terms:
[tex]\[
t + t^3 - t^2 - t^4
\][/tex]
Thus, the simplified expression for:
[tex]\[
\tan \theta \cdot (1 - \tan \theta) \cdot (1 + \tan^2 \theta)
\][/tex]
is:
[tex]\[
t (1 - t) (1 + t^2) = t \cdot (1 - t) \cdot (1 + t^2) = t (1 - t) (t^2 + 1)
\][/tex]
So, our final expression is:
[tex]\[
t (1 - t) (t^2 + 1)
\][/tex]
