Question

Simplify each expression. $\frac{\tan \theta \csc \theta}{\sec \theta}$

   Simplify each expression.
$\frac{\tan \theta \csc \theta}{\sec \theta}$

Algebra 2

Holliday, Luchin,… 1st Edition

Chapter 14, Problem 51

Instant Answer

Solved by Expert Elizabeth Xu

04/29/2020

Step 1

We have: \[\frac{\tan \theta \csc \theta}{\sec \theta} = \frac{\frac{\sin \theta}{\cos \theta} \cdot \frac{1}{\sin \theta}}{\frac{1}{\cos \theta}}\] Show more…

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Simplify each expression. $\frac{\tan \theta \csc \theta}{\sec \theta}$

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Key Concepts

Trigonometric Functions in Terms of Sine and Cosine

Many trigonometric expressions can be simplified by rewriting all functions in terms of sine and cosine. This approach relies on the fundamental definitions, such as tan(?) = sin(?)/cos(?), sec(?) = 1/cos(?), and csc(?) = 1/sin(?), which transforms complex trigonometric expressions into simpler algebraic forms.

Reciprocal Identities

Reciprocal identities define the relationships between trigonometric functions and their reciprocals. For instance, they establish that sec(?) is the reciprocal of cos(?) and csc(?) is the reciprocal of sin(?). These identities are useful for rewriting expressions and simplifying them through substitution.

Simplification of Complex Fractions

Simplifying complex fractions involves rewriting the expressions by combining like terms and canceling out common factors. By expressing trigonometric functions in their sine and cosine forms, one can often cancel terms and simplify the fraction to its most reduced form, as demonstrated in the problem where the given expression simplifies cleanly after such cancellation.

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