Solved on Feb 16, 2024

Find $\sec(-x)$ if $\sec(x)=2$ .

STEP 1

Assumptions
1. $\sec(x) = \frac{1}{\cos(x)}$
2. $\sec(-x) = \frac{1}{\cos(-x)}$
3. $\cos(-x) = \cos(x)$ (cosine is an even function)
4. Given that $\sec(x) = 2$ , we need to find $\sec(-x)$

STEP 2

Since $\sec(x) = 2$ , we can write it as:
$\sec(x) = \frac{1}{\cos(x)} = 2$

STEP 3

To find $\sec(-x)$ , we use the property of the cosine function being even, which means $\cos(x) = \cos(-x)$ .

STEP 4

Using the definition of secant in terms of cosine for $\sec(-x)$ :
$\sec(-x) = \frac{1}{\cos(-x)}$

STEP 5

Substitute $\cos(-x)$ with $\cos(x)$ due to the even property of cosine:
$\sec(-x) = \frac{1}{\cos(x)}$

STEP 6

Since we know $\sec(x) = 2$ , and $\sec(x) = \frac{1}{\cos(x)}$ , we can substitute $\frac{1}{\cos(x)}$ with 2:
$\sec(-x) = 2$
Thus, $\sec(-x) = 2$ .

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Find $\sec(-x)$ if $\sec(x)=2$ .

STEP 1

Assumptions
1. $\sec(x) = \frac{1}{\cos(x)}$
2. $\sec(-x) = \frac{1}{\cos(-x)}$
3. $\cos(-x) = \cos(x)$ (cosine is an even function)
4. Given that $\sec(x) = 2$ , we need to find $\sec(-x)$

STEP 2

Since $\sec(x) = 2$ , we can write it as:
$\sec(x) = \frac{1}{\cos(x)} = 2$

STEP 3

To find $\sec(-x)$ , we use the property of the cosine function being even, which means $\cos(x) = \cos(-x)$ .

STEP 4

Using the definition of secant in terms of cosine for $\sec(-x)$ :
$\sec(-x) = \frac{1}{\cos(-x)}$

STEP 5

Substitute $\cos(-x)$ with $\cos(x)$ due to the even property of cosine:
$\sec(-x) = \frac{1}{\cos(x)}$

STEP 6

Since we know $\sec(x) = 2$ , and $\sec(x) = \frac{1}{\cos(x)}$ , we can substitute $\frac{1}{\cos(x)}$ with 2:
$\sec(-x) = 2$
Thus, $\sec(-x) = 2$ .

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Find sec⁡(−x)\sec(-x)sec(−x) if sec⁡(x)=2\sec(x)=2sec(x)=2.

STEP 1

STEP 2

Since sec⁡(x)=2\sec(x) = 2sec(x)=2, we can write it as:sec⁡(x)=1cos⁡(x)=2\sec(x) = \frac{1}{\cos(x)} = 2sec(x)=cos(x)1​=2

STEP 3

To find sec⁡(−x)\sec(-x)sec(−x), we use the property of the cosine function being even, which means cos⁡(x)=cos⁡(−x)\cos(x) = \cos(-x)cos(x)=cos(−x).

STEP 4

Using the definition of secant in terms of cosine for sec⁡(−x)\sec(-x)sec(−x):sec⁡(−x)=1cos⁡(−x)\sec(-x) = \frac{1}{\cos(-x)}sec(−x)=cos(−x)1​

STEP 5

Substitute cos⁡(−x)\cos(-x)cos(−x) with cos⁡(x)\cos(x)cos(x) due to the even property of cosine:sec⁡(−x)=1cos⁡(x)\sec(-x) = \frac{1}{\cos(x)}sec(−x)=cos(x)1​

STEP 6

Since we know sec⁡(x)=2\sec(x) = 2sec(x)=2, and sec⁡(x)=1cos⁡(x)\sec(x) = \frac{1}{\cos(x)}sec(x)=cos(x)1​, we can substitute 1cos⁡(x)\frac{1}{\cos(x)}cos(x)1​ with 2:sec⁡(−x)=2\sec(-x) = 2sec(−x)=2Thus, sec⁡(−x)=2\sec(-x) = 2sec(−x)=2.

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Find $\sec(-x)$ if $\sec(x)=2$ .

Since $\sec(x) = 2$ , we can write it as:
$\sec(x) = \frac{1}{\cos(x)} = 2$

To find $\sec(-x)$ , we use the property of the cosine function being even, which means $\cos(x) = \cos(-x)$ .

Using the definition of secant in terms of cosine for $\sec(-x)$ :
$\sec(-x) = \frac{1}{\cos(-x)}$

Substitute $\cos(-x)$ with $\cos(x)$ due to the even property of cosine:
$\sec(-x) = \frac{1}{\cos(x)}$

Since we know $\sec(x) = 2$ , and $\sec(x) = \frac{1}{\cos(x)}$ , we can substitute $\frac{1}{\cos(x)}$ with 2:
$\sec(-x) = 2$
Thus, $\sec(-x) = 2$ .