Chapter 5.1, Problem 37E

To determine

To determine: The simplified form of the expression using fundamental identities.

Answer to Problem 37E

The required expression is $\sec \beta$ .

Explanation of Solution

Given information:

The trigonometric expression: $\sin \beta \tan \beta +\cos \beta$

Formula used:

Trigonometric fundamental identities:

  $\tan x=\frac{\sin x}{\cos x}$

  ${\sin }^{2}x+{\cos }^{2}x=1$

  $\frac{1}{\cos x}=\sec x$

Calculation:

The given trigonometric expression is $\sin \beta \tan \beta +\cos \beta$ .

Rewrite the above expression using the formula $\tan x=\frac{\sin x}{\cos x}$ as:

  $\begin{array}{c}\sin \beta \tan \beta +\cos \beta =\sin \beta \frac{\sin \beta }{\cos \beta }+\cos \beta \\ =\frac{{\sin }^2\beta +{\cos }^2\beta }{\cos \beta }\end{array}$

To simplify the above expression, use the formulas ${\sin }^{2}x+{\cos }^{2}x=1$ and $\frac{1}{\cos x}=\sec x$ .

  $\begin{array}{c}\frac{{\sin }^2\beta +{\cos }^2\beta }{\cos \beta }=\frac{1}{\cos \beta }\\ =\sec \beta \end{array}$

Hence, the required expression is $\sec \beta$ .