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Chapter 5: Problem 37

Use the fundamental identities to simplify the expression. There is more than one correct form of each answer. $$\cot \theta \sec \theta$$

Short Answer

Expert verified
The simplified expression for \( \cot \theta \sec \theta \) is \( \frac{1}{\sin \theta} \), or, in other words, \( \csc \theta \).

Step by step solution

01

Identify and rewrite cotangent and secant in terms of sine and cosine

\(\cot \theta\) can be rewritten as \(\frac{1}{\tan \theta}\) and \(\tan \theta\) is \(\frac{\sin \theta}{\cos \theta}\). So rewriting \(\cot \theta\) gives us \(\frac{\cos \theta}{\sin \theta}\). Similarly \(\sec \theta\) can be rewritten as \(\frac{1}{\cos \theta}\). Now replace \(\cot \theta\) and \(\sec \theta\) in the original equation with \(\frac{\cos \theta}{\sin \theta}\) and \(\frac{1}{\cos \theta}\) respectively.
02

Multiply the two fractions

To multiply two fractions, multiply the numerators together and the denominators together. In this case, that gives us \(\frac{\cos \theta}{\sin \theta} \cdot \frac{1}{\cos \theta} = \frac{\cos \theta}{\sin \theta \cos \theta}\).
03

Simplify the Fraction

Finally, simplify the fraction by cancelling out the common factor in the numerator and the denominator. This gives us \(\frac{1}{\sin \theta}\).

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Most popular questions from this chapter

A Ferris wheel is built such that the height \(h\) (in feet) above ground of a seat on the wheel at time \(t\) (in minutes) can be modeled by \(h(t)=53+50 \sin \left(\frac{\pi}{16} t-\frac{\pi}{2}\right)\) The wheel makes one revolution every 32 seconds. The ride begins when \(t=0\). (a) During the first 32 seconds of the ride, when will a person on the Ferris wheel be 53 feet above ground? (b) When will a person be at the top of the Ferris wheel for the first time during the ride? If the ride lasts 160 seconds, how many times will a person be at the top of the ride, and at what times?

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