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Chapter 5: Problem 54

Use the fundamental identities to simplify the expression. There is more than one correct form of each answer. $$\cos t\left(1+\tan ^{2} t\right)$$

Short Answer

Expert verified
The simplified form of the expression \(\cos t\left(1+\tan ^{2} t\right)\) is \(\sec t\).

Step by step solution

01

Identify the Identity to Use

In this case, observe that the expression has a term of the form \(1+\tan ^{2} t\) which directly corresponds to \(\sec^2 t\) from the Pythagorean identity. This allows us to apply this identity to simplify the expression.
02

Apply the Pythagorean Identity

Replace \(1 + \tan^2 t\) with \(\sec^2 t\). Doing so transforms the original expression \(\cos t \left(1 + \tan^2 t\right)\) into \(\cos t \cdot \sec^2 t\).
03

Further Simplify the Expression

\(\sec t\) is the reciprocal of \(\cos t\), i.e. \(\sec t = \frac{1}{\cos t}\). Therefore \(\sec^2 t = \frac{1}{\cos^2 t}\). Substitute this into the expression to get \(\cos t \cdot \frac{1}{\cos^2 t} = \frac{1}{\cos t}\) which is equal to \(\sec t\).

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Most popular questions from this chapter

Use inverse functions where needed to find all solutions of the equation in the interval \([0,2 \pi)\). $$\sec ^{2} x+\tan x-3=0$$

Use a graphing utility to approximate the solutions (to three decimal places) of the equation in the given interval. $$\cos ^{2} x-2 \cos x-1=0, \quad[0, \pi]$$

Use inverse functions where needed to find all solutions of the equation in the interval \([0,2 \pi)\). $$2 \sin ^{2} x-7 \sin x+3=0$$

The table shows the average daily high temperatures in Houston \(H\) (in degrees Fahrenheit) for month \(t,\) with \(t=1\) corresponding to January. (Source: National Climatic Data Center) $$ \begin{array}{|c|c|} \hline \text { Month, } t & \text { Houston, } \boldsymbol{H} \\ \hline 1 & 62.3 \\ 2 & 66.5 \\ 3 & 73.3 \\ 4 & 79.1 \\ 5 & 85.5 \\ 6 & 90.7 \\ 7 & 93.6 \\ 8 & 93.5 \\ 9 & 89.3 \\ 10 & 82.0 \\ 11 & 72.0 \\ 12 & 64.6 \\ \hline \end{array} $$ (a) Create a scatter plot of the data. (b) Find a cosine model for the temperatures in Houston. (c) Use a graphing utility to graph the data points and the model for the temperatures in Houston. How well does the model fit the data? (d) What is the overall average daily high temperature in Houston? (e) Use a graphing utility to describe the months during which the average daily high temperature is above \(86^{\circ} \mathrm{F}\) and below \(86^{\circ} \mathrm{F}\).

Determine whether the statement is true or false. Justify your answer. The equation \(2 \sin 4 t-1=0\) has four times the number of solutions in the interval \([0,2 \pi)\) as the equation \(2 \sin t-1=0\).
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