To determine whether the statement [tex]\(\tan \theta \cdot \cos \theta = \sin \theta\)[/tex] is true for any [tex]\(\theta \neq (2k+1)\frac{\pi}{2}\)[/tex], we begin by analyzing the given trigonometric identity.
1. Understand the Components:
- Recall that [tex]\(\tan \theta\)[/tex] (tangent of [tex]\(\theta\)[/tex]) is defined as [tex]\(\frac{\sin \theta}{\cos \theta}\)[/tex].
- Also, we know that [tex]\(\sin \theta\)[/tex] (sine of [tex]\(\theta\)[/tex]) and [tex]\(\cos \theta\)[/tex] (cosine of [tex]\(\theta\)[/tex]) are fundamental trigonometric functions.
2. Set Up the Equation:
- We are given [tex]\(\tan \theta \cdot \cos \theta\)[/tex].
- Substitute [tex]\(\tan \theta\)[/tex] using the trigonometric definition: [tex]\(\tan \theta = \frac{\sin \theta}{\cos \theta}\)[/tex].
3. Simplify the Expression:
[tex]\[
\tan \theta \cdot \cos \theta = \left( \frac{\sin \theta}{\cos \theta} \right) \cdot \cos \theta
\][/tex]
4. Cancel Out Common Terms:
- In the expression [tex]\(\left( \frac{\sin \theta}{\cos \theta} \right) \cdot \cos \theta\)[/tex], the [tex]\(\cos \theta\)[/tex] in the numerator and denominator cancel each other out:
[tex]\[
\left( \frac{\sin \theta}{\cos \theta} \right) \cdot \cos \theta = \sin \theta
\][/tex]
5. Evaluate the Validity:
- The simplified form confirms that [tex]\(\tan \theta \cdot \cos \theta = \sin \theta\)[/tex].
- However, this simplification holds as long as [tex]\(\cos \theta \neq 0\)[/tex].
6. Determine the Restrictions:
- [tex]\(\cos \theta = 0\)[/tex] at specific angles where [tex]\( \theta = (2k+1)\frac{\pi}{2} \)[/tex] for any integer [tex]\(k\)[/tex]. At these angles, [tex]\(\tan \theta\)[/tex] is undefined because it involves division by zero.
7. Conclusion:
- Therefore, the original statement [tex]\(\tan \theta \cdot \cos \theta = \sin \theta\)[/tex] is true for any [tex]\(\theta \neq (2k+1)\frac{\pi}{2}\)[/tex].
In sum, the statement is true for [tex]\(\theta \neq (2k+1)\frac{\pi}{2}\)[/tex].
