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Chapter 5: Problem 43

Use the fundamental identities to simplify the expression. There is more than one correct form of each answer. $$\sin \beta \tan \beta+\cos \beta$$

Short Answer

Expert verified
The simplified form of the expression is \(\sec \beta\).

Step by step solution

01

Replace tan with sin/cos

To simplify the given expression, recall that the tangent of an angle in trigonometry is the ratio of the sine to the cosine. This can be expressed as \(\tan \beta = \frac{\sin \beta}{\cos \beta}\). So, the given expression \(\sin \beta \tan \beta+\cos \beta\) can be rewritten as \(\sin \beta \cdot \frac{\sin \beta}{\cos \beta} + \cos \beta\)
02

Simplify the Expression

Simplify the first term of the revised expression by cancelling out the common factor of \(\sin \beta\) in the numerator and denominator to get \(\frac{\sin^2 \beta}{\cos \beta} + \cos \beta\)
03

Convert to Common Denominator

Since the two terms in the expression have different denominators, convert to a common denominator in order to simplify further. The common denominator is \(\cos \beta\), so the expression becomes \(\frac{\sin^2 \beta + \cos^2 \beta}{\cos \beta}\)
04

Apply Pythagorean Identity

The numerator of the fraction is the Pythagorean identity \( \sin^2 \beta + \cos^2 \beta \), which equals 1. Hence, the expression simplifies to \(\frac{1}{\cos \beta}\) which is equivalent to \(\sec \beta\).

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