Question

Simplify each expression. $$ \frac{\sin \theta+\tan \theta}{1+\cos \theta} $$

   Simplify each expression.
$$
\frac{\sin \theta+\tan \theta}{1+\cos \theta}
$$

Algebra 2

Prentice Hall, Basia… 1st Edition

Chapter 14, Problem 87

Instant Answer

Solved by Expert Wayne Loutet

02/02/2020

Step 1

The sine function remains the same, and the tangent function can be written as the ratio of sine to cosine. So, the expression becomes: $$ \frac{\sin \theta+\frac{\sin \theta}{\cos \theta}}{1+\cos \theta} $$ Show more…

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Simplify each expression. $$ \frac{\sin \theta+\tan \theta}{1+\cos \theta} $$

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Key Concepts

Factorization and Cancellation

Identifying and factoring out common factors in both the numerator and the denominator allows for canceling, which simplifies the overall expression. This technique is a key step in reducing complex trigonometric ratios to simpler forms.

Common Denominator Technique

Combining fractional expressions by writing them with a common denominator is a standard algebraic method. This technique is especially useful in trigonometry for merging terms and revealing common factors that can subsequently be canceled to simplify the expression.

Rewriting in Terms of Sine and Cosine

Rewriting trigonometric functions in terms of sine and cosine can simplify the process of combining and manipulating them. Expressing tan(?) as sin(?)/cos(?) is an example of this, as it allows disparate terms to be combined over a common denominator, facilitating further simplification.

Trigonometric Identities

Trigonometric identities such as tan(?) = sin(?)/cos(?) are fundamental tools that allow one to rewrite and simplify expressions involving sine, cosine, and tangent. These identities help in expressing trigonometric functions in different forms and play a crucial role in simplifying complex trigonometric expressions.

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