QuestionRewrite $\cos x^{2} \cot x + \sin x$ using $\sin x$ and $\cos x$ , then simplify.

Studdy Solution

STEP 1

Assumptions1. We are given the expression $\cos x^{} \cot x+\sin x$ . . We need to rewrite this expression in terms of $\sin x$ and $\cos x$ .
3. We can use the trigonometric identity $\sin x+\cos x=1$ to simplify the expression.
4. We know that $\cot x = \frac{\cos x}{\sin x}$ .

STEP 2

First, we rewrite $\cot x$ in terms of $\sin x$ and $\cos x$ .
$\cot x = \frac{\cos x}{\sin x}$

STEP 3

Now, substitute $\cot x$ in the given expression with $\frac{\cos x}{\sin x}$ .
$\cos x^{2} \cot x+\sin x = \cos x^{2} \frac{\cos x}{\sin x}+\sin x$

STEP 4

implify the expression by multiplying $\cos x^{2}$ and $\frac{\cos x}{\sin x}$ .
$\cos x^{2} \frac{\cos x}{\sin x}+\sin x = \frac{\cos^{3} x}{\sin x}+\sin x$

STEP 5

To add these two terms, we need to get a common denominator. The common denominator here is $\sin x$ .
$\frac{\cos^{3} x}{\sin x}+\sin x = \frac{\cos^{3} x + \sin^{2} x}{\sin x}$

STEP 6

Next, we use the identity $\sin^{2} x+\cos^{2} x=1$ to simplify the numerator.
$\frac{\cos^{3} x + \sin^{2} x}{\sin x} = \frac{\cos^{3} x + (1-\cos^{2} x)}{\sin x}$

STEP 7

implify the numerator.
$\frac{\cos^{3} x +1-\cos^{2} x}{\sin x} = \frac{1+\cos x(\cos^{2} x -1)}{\sin x}$

STEP 8

Factor out $\cos x$ in the numerator.
$\frac{1+\cos x(\cos^{2} x -1)}{\sin x} = \frac{1+\cos x(\cos x -1)(\cos x +1)}{\sin x}$

STEP 9

We can't simplify the expression further, so this is our final answer.
$\frac{+\cos x(\cos x -)(\cos x +)}{\sin x}$ So, $\cos x^{2} \cot x+\sin x$ can be written as $\frac{+\cos x(\cos x -)(\cos x +)}{\sin x}$ in terms of $\sin x$ and $\cos x$ .

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