Solved on Jan 11, 2024

Find the derivative of $\sec x \cos x$ .

STEP 1

Assumptions
1. We are working with trigonometric functions.
2. The variable $x$ represents an angle in radians or degrees.
3. The goal is to simplify the expression $\sec x \cos x$ .

STEP 2

Recall the definition of the secant function in terms of the cosine function.
$\sec x = \frac{1}{\cos x}$

STEP 3

Substitute the definition of $\sec x$ into the expression $\sec x \cos x$ .
$\sec x \cos x = \left(\frac{1}{\cos x}\right) \cos x$

STEP 4

Simplify the expression by multiplying $\frac{1}{\cos x}$ by $\cos x$ .
$\left(\frac{1}{\cos x}\right) \cos x = \frac{\cos x}{\cos x}$

STEP 5

Recognize that $\frac{\cos x}{\cos x}$ simplifies to 1, since any nonzero number divided by itself equals 1.
$\frac{\cos x}{\cos x} = 1$

STEP 6

Conclude that the simplified form of the expression $\sec x \cos x$ is 1.
$\sec x \cos x = 1$
The simplified expression is 1.

Was this helpful?

Find the derivative of $\sec x \cos x$ .

STEP 1

Assumptions
1. We are working with trigonometric functions.
2. The variable $x$ represents an angle in radians or degrees.
3. The goal is to simplify the expression $\sec x \cos x$ .

STEP 2

Recall the definition of the secant function in terms of the cosine function.
$\sec x = \frac{1}{\cos x}$

STEP 3

Substitute the definition of $\sec x$ into the expression $\sec x \cos x$ .
$\sec x \cos x = \left(\frac{1}{\cos x}\right) \cos x$

STEP 4

Simplify the expression by multiplying $\frac{1}{\cos x}$ by $\cos x$ .
$\left(\frac{1}{\cos x}\right) \cos x = \frac{\cos x}{\cos x}$

STEP 5

Recognize that $\frac{\cos x}{\cos x}$ simplifies to 1, since any nonzero number divided by itself equals 1.
$\frac{\cos x}{\cos x} = 1$

STEP 6

Conclude that the simplified form of the expression $\sec x \cos x$ is 1.
$\sec x \cos x = 1$
The simplified expression is 1.

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Find the derivative of sec⁡xcos⁡x\sec x \cos xsecxcosx.

STEP 1

Assumptions1. We are working with trigonometric functions.2. The variable x x x represents an angle in radians or degrees.3. The goal is to simplify the expression sec⁡xcos⁡x \sec x \cos x secxcosx.

STEP 2

Recall the definition of the secant function in terms of the cosine function.sec⁡x=1cos⁡x\sec x = \frac{1}{\cos x}secx=cosx1​

STEP 3

Substitute the definition of sec⁡x \sec x secx into the expression sec⁡xcos⁡x \sec x \cos x secxcosx.sec⁡xcos⁡x=(1cos⁡x)cos⁡x\sec x \cos x = \left(\frac{1}{\cos x}\right) \cos xsecxcosx=(cosx1​)cosx

STEP 4

Simplify the expression by multiplying 1cos⁡x \frac{1}{\cos x} cosx1​ by cos⁡x \cos x cosx.(1cos⁡x)cos⁡x=cos⁡xcos⁡x\left(\frac{1}{\cos x}\right) \cos x = \frac{\cos x}{\cos x}(cosx1​)cosx=cosxcosx​

STEP 5

Recognize that cos⁡xcos⁡x \frac{\cos x}{\cos x} cosxcosx​ simplifies to 1, since any nonzero number divided by itself equals 1.cos⁡xcos⁡x=1\frac{\cos x}{\cos x} = 1cosxcosx​=1

STEP 6

Conclude that the simplified form of the expression sec⁡xcos⁡x \sec x \cos x secxcosx is 1.sec⁡xcos⁡x=1\sec x \cos x = 1secxcosx=1The simplified expression is 1.

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Find the derivative of $\sec x \cos x$ .

Assumptions
1. We are working with trigonometric functions.
2. The variable $x$ represents an angle in radians or degrees.
3. The goal is to simplify the expression $\sec x \cos x$ .

Recall the definition of the secant function in terms of the cosine function.
$\sec x = \frac{1}{\cos x}$

Substitute the definition of $\sec x$ into the expression $\sec x \cos x$ .
$\sec x \cos x = \left(\frac{1}{\cos x}\right) \cos x$

Simplify the expression by multiplying $\frac{1}{\cos x}$ by $\cos x$ .
$\left(\frac{1}{\cos x}\right) \cos x = \frac{\cos x}{\cos x}$

Recognize that $\frac{\cos x}{\cos x}$ simplifies to 1, since any nonzero number divided by itself equals 1.
$\frac{\cos x}{\cos x} = 1$

Conclude that the simplified form of the expression $\sec x \cos x$ is 1.
$\sec x \cos x = 1$
The simplified expression is 1.