CONCEPT PREVIEW Fill in the blank(s) to correctly complete each sentence. Given tan θ = 1/cot θ , two equivalent forms of this identity are cot θ = 1/______ and tan θ . ______ = 1 .

Verified step by step guidance

1. Recall the definitions of the tangent and cotangent functions: \(\tan \theta = \frac{\sin \theta}{\cos \theta}\) and \(\cot \theta = \frac{\cos \theta}{\sin \theta}\).
2. Given the identity \(\tan \theta = \frac{1}{\cot \theta}\), rewrite \(\cot \theta\) in terms of \(\tan \theta\) by taking the reciprocal of both sides, which gives \(\cot \theta = \frac{1}{\tan \theta}\).
3. The first blank corresponds to the expression for \(\cot \theta\) in terms of \(\tan \theta\), so fill it with \(\tan \theta\).
4. For the second blank, consider the product \(\tan \theta \cdot \cot \theta\). Using the reciprocal relationship, this product equals 1, so fill the blank with \(\cot \theta\).
5. Summarize the equivalent forms: \(\cot \theta = \frac{1}{\tan \theta}\) and \(\tan \theta \cdot \cot \theta = 1\).

Key Concepts

Reciprocal Identities

Reciprocal identities in trigonometry express the relationship between pairs of functions such as tangent and cotangent. Specifically, tan θ is the reciprocal of cot θ, meaning tan θ = 1/cot θ and cot θ = 1/tan θ. Understanding these helps in rewriting and simplifying trigonometric expressions.

Definition of Tangent and Cotangent

Tangent of an angle θ is defined as the ratio of the sine to the cosine of θ (tan θ = sin θ / cos θ), while cotangent is the inverse ratio (cot θ = cos θ / sin θ). Recognizing these definitions clarifies why tan θ and cot θ are reciprocals.

Multiplicative Inverse Property

The multiplicative inverse property states that a number multiplied by its reciprocal equals one. Applying this to trigonometric functions, tan θ multiplied by cot θ equals 1, reinforcing the identity tan θ * cot θ = 1.