By applying algebraic properties and trigonometric definitions we conclude that the trigonometric expression [tex]\frac{\sin \theta}{1 - \cos \theta} + \frac{\tan \theta}{1 + \cos \theta}[/tex] is equivalent to [tex]\cot \theta + \sec \theta \cdot \csc \theta[/tex].
How to demonstrate a trigonometric expression
In this question we need to apply algebraic properties and trigonometric definitions to prove that an equivalence between two trigonometric expressions. Now we proceed to show the required procedure:
- [tex]\frac{\sin \theta}{1 - \cos \theta} + \frac{\tan \theta}{1 + \cos \theta}[/tex] Given
- [tex]\frac{\sin \theta}{1 - \cos \theta} + \frac{\sin \theta}{\cos \theta \cdot (1 + \cos \theta)}[/tex] tan θ = sin θ/cos θ
- [tex]\sin \theta \cdot \left[\frac{1}{1-\cos \theta} + \frac{1}{\cos \theta \cdot (1+\cos \theta)} \right][/tex] Distributive property
- [tex]\sin \theta \cdot \left[\frac{\cos \theta \cdot (1 + \cos \theta) + (1 - \cos \theta)}{(1-\cos \theta)\cdot (1 + \cos \theta)\cdot \cos \theta} \right][/tex] [tex]\frac{x}{y} + \frac{w}{z} = \frac{x\cdot z + y\cdot w}{y\cdot z}[/tex]
- [tex]\sin \theta \cdot \left[\frac{\cos\theta + \cos^{2}\theta + 1 - \cos \theta}{\cos \theta \cdot (1 - \cos^{2} \theta)}\right][/tex] (a + b) · (a + b) = a² - b²/Commutative property/Distributive property/Definition of power
- [tex]\sin \theta \cdot \left[\frac{\cos^{2}\theta + 1}{\cos \theta\cdot \sin^{2}\theta}\right][/tex] Commutative and associative properties/Existence of the additive inverse/Modulative property
- [tex]\frac{\sin \theta \cdot \cos^{2}\theta + \sin \theta}{\cos \theta \cdot \sin^{2}\theta}[/tex] Distributive property
- [tex]\frac{\sin \theta \cdot \cos^{2}\theta}{\cos \theta \cdot \sin^{2}\theta} + \frac{\sin \theta}{\cos \theta\cdot \sin^{2}\theta}[/tex] [tex]\frac{x+y}{z} = \frac{x}{z} + \frac{y}{z}[/tex]
- [tex]\frac{\cos \theta}{\sin \theta} + \left(\frac{1}{\cos \theta}\right)\cdot (\frac{1}{\sin \theta} )[/tex] Definition of power/Associative property/Existence of multiplicative inverse/Modulative property
- [tex]\cot \theta + \sec \theta \cdot \csc \theta[/tex] [tex]\cot \theta = \frac{\cos \theta}{\sin \theta}[/tex], [tex]\sec \theta = \frac{1}{\cos \theta}[/tex], [tex]\csc \theta = \frac{1}{\sin \theta}[/tex]/Result
By applying algebraic properties and trigonometric definitions we conclude that the trigonometric expression [tex]\frac{\sin \theta}{1 - \cos \theta} + \frac{\tan \theta}{1 + \cos \theta}[/tex] is equivalent to [tex]\cot \theta + \sec \theta \cdot \csc \theta[/tex].
To learn more on trigonometric expressions: https://brainly.com/question/10083069
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