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Chapter 16: Problem 11

The expression \(\tan \theta \sec \theta\left(1-\sin ^{2} \theta\right) / \cos \theta \operatorname{simplifies~to~}\) a) \(\sin \theta\) b) \(\cos \theta\) c) \(\tan \theta\) d) \(\sec \theta\)

Short Answer

Expert verified
Question: Simplify the expression \(\frac{\tan \theta \sec \theta\left(1-\sin ^{2} \theta\right)}{\cos \theta}\). Answer: \(\tan \theta\)

Step by step solution

01

Write down the given expression

The given expression is: $$ \frac{\tan \theta \sec \theta\left(1-\sin ^{2} \theta\right)}{\cos \theta} $$
02

Replace the tan and sec functions using their definition

We know that \(\tan \theta = \frac{\sin \theta}{\cos \theta}\) and \(\sec \theta = \frac{1}{\cos \theta}\), so we can rewrite the expression as: $$ \frac{\frac{\sin \theta}{\cos \theta} \cdot \frac{1}{\cos \theta}\left(1-\sin ^{2} \theta\right)}{\cos \theta} $$
03

Simplify the fraction

Multiply the numerators, which results in: $$ \frac{\sin \theta (1-\sin ^{2} \theta)}{\cos ^{3} \theta} $$
04

Use the Pythagorean identity

We know that \(\sin^2\theta + \cos^2\theta = 1\), so \(1 - \sin^2\theta = \cos^2\theta\). Replace in the expression: $$ \frac{\sin \theta \cdot \cos^2 \theta}{\cos ^{3} \theta} $$
05

Simplify the expression

Cancel \(\cos^2\theta\) from the numerator and denominator, leaving us with: $$ \frac{\sin \theta}{\cos \theta} $$
06

Rewrite the expression using trigonometric functions

We can rewrite the expression back to the tangent function: $$ \tan \theta $$ The simplified expression is \(\tan \theta\), which corresponds to option c) in the exercise.

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Most popular questions from this chapter

The \(x\) - and \(y\)-axes are the asymptotes of a hyperbola that passes through the point \((2,2)\). Its equation is a) \(x^{2}-y^{2}=0\) b) \(x y=4\) c) \(y^{2}-x^{2}=0\) d) \(x^{2}+y^{2}=4\)

Determine \(\left[\begin{array}{ll}1 & 2 \\ 2 & 1\end{array}\right]\left[\begin{array}{rr}-1 & 0 \\ 1 & 2\end{array}\right]\). a) \(\left[\begin{array}{rr}1 & 4 \\ -1 & 2\end{array}\right]\) b) \(\left[\begin{array}{rr}1 & -1 \\ 4 & 2\end{array}\right]\) c) \(\left[\begin{array}{r}1 \\ -1\end{array}\right]\) d) \(\left[\begin{array}{l}4 \\ 2\end{array}\right]\)

Derive an expression \(\int x \cos x d x\) a) \(x \cos x-\sin x+C\) c) \(x \sin x-\cos x+C\) b) \(x \sin x+\cos x+C\) d) \(x \cos x+\sin x+C\)

If \(y=\ln x+e^{x} \sin x\), find \(d y / d x\) at \(x=1\). a) \(1.23\) b) \(3.68\) c) \(4.76\) d) \(6.12\)

Solve for \(\left[x_{i}\right]\) $$ \begin{aligned} 3 x_{1}+2 x_{2} &=-2 \\ x_{1}-x_{2}+x_{3} &=0 \\ 4 x_{1}+2 x_{3} &=4 \end{aligned} $$ a) \(\left[\begin{array}{r}2 \\ 4 \\ -6\end{array}\right]\) \(\left[\begin{array}{l}2 \\ 4 \\ 8\end{array}\right]\) d) \(\left[\begin{array}{r}8 \\ 14\end{array}\right]\)
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