Chapter 10.4, Problem 7MRPS

Solution

a.

To show: That the expression $\csc x\cot x-\cos x$ for the rate of can be can be written as $\cos x{\cot }^{2}x$ .

Given information:

The given expression is: $\csc x\cot x-\cos x$

Explanation:

Consider the given expression: $\csc x\cot x-\cos x$

  $\csc x\cot x-\cos x$

Substitute $\csc x=\frac{1}{\sin x}\text{ and }\cot x=\frac{\cos x}{\sin x}$ in the expression,

  $\begin{array}{c}\csc x\cot x-\cos x=\frac{1}{\sin x}\cdot \frac{\cos x}{\sin x}-\cos x\\ =\frac{\cos x}{{\sin }^{2}x}-\cos x\\ =\cos x\left(\frac{1}{{\sin }^{2}x}-1\right)\\ =\cos x\left(\frac{1-{\sin }^{2}x}{{\sin }^{2}x}\right)\end{array}$

Now substitute $1-{\sin }^{2}x={\cos }^{2}x$ ,

  $\cos x\left(\frac{1-{\sin }^{2}x}{{\sin }^{2}x}\right)=\cos x\left(\frac{{\cos }^{2}x}{{\sin }^{2}x}\right)$

Now $\frac{{\cos }^{2}x}{{\sin }^{2}x}$ can be written as ${\cot }^{2}x$ ,

  $\cos x\left(\frac{{\cos }^{2}x}{{\sin }^{2}x}\right)=\cos x{\cot }^{2}x$

Therefore, it can be said that $\csc x\cot x-\cos x=\cos x{\cot }^{2}x$ .

b.

To state: The identities used to solve part (a).

Reciprocal identity, Quotient identity, and Pythagorean identity are used in part (a).

Given information:

The given statement says to list the identities used in part (a).

Explanation:

The identities used in part (a) are:

Reciprocal identity: $\csc x=\frac{1}{\sin x}$

Quotient identity: $\cot x=\frac{\cos x}{\sin x}$

Pythagorean identity: ${\sin }^2\theta +{\cos }^2\theta =1$