QuestionRewrite and simplify $\cos x^{*} \cot x + \sin x$ using $\sin x$ and $\cos x$ .

Studdy Solution

STEP 1

Assumptions1. We are given the expression $\cos x^{*} \cot x+\sin x$ . . We have to rewrite this expression in terms of $\sin x$ and $\cos x$ .
3. We can use the trigonometric identities to simplify the expression.

STEP 2

First, we need to rewrite $\cot x$ in terms of $\sin x$ and $\cos x$ . The cotangent of $x$ , $\cot x$ , is the reciprocal of the tangent of $x$ , $\tan x$ . And $\tan x$ is equal to $\frac{\sin x}{\cos x}$ . Therefore, $\cot x$ is equal to $\frac{\cos x}{\sin x}$ .
$\cot x = \frac{\cos x}{\sin x}$

STEP 3

Now, substitute $\cot x$ in the original expression with $\frac{\cos x}{\sin x}$ .
$\cos x^{} \cot x+\sin x = \cos x^{} \left(\frac{\cos x}{\sin x}\right) + \sin x$

STEP 4

implify the expression by distributing $\cos x$ .
$\cos x^{*} \left(\frac{\cos x}{\sin x}\right) + \sin x = \frac{\cos^2 x}{\sin x} + \sin x$

STEP 5

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

STEP 6

Now, use the identity $\sin^2 x + \cos^2 x =1$ to simplify the numerator.
$\frac{\cos^2 x + \sin^2 x}{\sin x} = \frac{1}{\sin x}$

STEP 7

Finally, $\frac{1}{\sin x}$ is equal to $\csc x$ .
$\frac{1}{\sin x} = \csc x$ So, the expression $\cos x^{*} \cot x+\sin x$ simplifies to $\csc x$ .

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