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swat32
2 years ago
10

Cleft%28x%5Cright%29%7D%7B%5Ccsc%5Cleft%28x%5Cright%29%5Ccos%5E%7B2%7D%5Cleft%28x%5Cright%29%7D" id="TexFormula1" title="\frac{\sec\left(x\right)}{\cos\left(x\right)}-\frac{\sin\left(x\right)}{\csc\left(x\right)\cos^{2}\left(x\right)}" alt="\frac{\sec\left(x\right)}{\cos\left(x\right)}-\frac{\sin\left(x\right)}{\csc\left(x\right)\cos^{2}\left(x\right)}" align="absmiddle" class="latex-formula">Use the basic identities to change the expression to one involving only sines and cosines. Then simplify to a basic trig function.
Mathematics
1 answer:
DanielleElmas [232]2 years ago
7 0

Answer:

1

Step-by-step explanation:

First, convert all the secants and cosecants to cosine and sine, respectively. Recall that csc(x)=1/sin(x) and sec(x)=1/cos(x).

Thus:

\frac{sec(x)}{cos(x)} -\frac{sin(x)}{csc(x)cos^2(x)}

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

Let's do the first part first: (Recall how to divide fractions)

\frac{\frac{1}{cos(x)} }{cos(x)}=\frac{1}{cos(x)} \cdot \frac{1}{cos(x)}=\frac{1}{cos^2(x)}

For the second term:

\frac{sin(x)}{\frac{cos^2(x)}{sin(x)} } =\frac{sin(x)}{1} \cdot\frac{sin(x)}{cos^2(x)}=\frac{sin^2(x)}{cos^2(x)}

So, all together: (same denominator; combine terms)

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

Note the numerator; it can be derived from the Pythagorean Identity:

sin^2(x)+cos^2(x)=1; cos^2(x)=1-sin^2(x)

Thus, we can substitute the numerator:

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

Everything simplifies to 1.

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Step-by-step explanation:

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2 years ago
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Jenny multiplies the square root of her favorite positive integer by $\sqrt{2}$. Her product is an integer. a) Name three number
Dmitry [639]

Answer:

Part a) 2,50,18

Part b) When Jenny divides the square root of her favorite positive integer by \sqrt{2}, she gets an integer

Step-by-step explanation:

Let

x-------> the favorite positive integer

Part a)

1) For x=2

\sqrt{2}*\sqrt{2}=\sqrt{4}=2 -----> the product is an integer

so

The number x=2 could be Jenny favorite positive integer

2) For x=50

\sqrt{50}*\sqrt{2}=\sqrt{100}=10 -----> the product is an integer

so

The number x=50 could be Jenny favorite positive integer

3) For x=18

\sqrt{18}*\sqrt{2}=\sqrt{36}=6 -----> the product is an integer

so

The number x=18 could be Jenny favorite positive integer

Part B)

1) For x=2

\sqrt{2}/\sqrt{2}=\sqrt{1}=1 -----> the result is an integer

2) For x=50

\sqrt{50}/\sqrt{2}=\sqrt{25}=5 -----> the result is an integer

3) For x=18

\sqrt{18}/\sqrt{2}=\sqrt{9}=3 -----> the result is an integer

Therefore

When Jenny divides the square root of her favorite positive integer by \sqrt{2} , she gets an integer

4 0
3 years ago
Andrew's mother reasoned that Chris probably took out a 30-year mortgage. Thirty years is a popular mortgage length because paym
melisa1 [442]
They would have increased because the 20% is bigger than the 5% and 5%=monthly housing expenses raising slow. 20%=monthly housing expenses raising higher.
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3 years ago
7x-11=5(x-2)+2x-1= ?​
yan [13]
0 = 0


Simplifying
7x + -11 = 5(x + -2) + 2x + -1

Reorder the terms:
-11 + 7x = 5(x + -2) + 2x + -1

Reorder the terms:
-11 + 7x = 5(-2 + x) + 2x + -1
-11 + 7x = (-2 * 5 + x * 5) + 2x + -1
-11 + 7x = (-10 + 5x) + 2x + -1

Reorder the terms:
-11 + 7x = -10 + -1 + 5x + 2x

Combine like terms: -10 + -1 = -11
-11 + 7x = -11 + 5x + 2x

Combine like terms: 5x + 2x = 7x
-11 + 7x = -11 + 7x

Add '11' to each side of the equation.
-11 + 11 + 7x = -11 + 11 + 7x

Combine like terms: -11 + 11 = 0
0 + 7x = -11 + 11 + 7x
7x = -11 + 11 + 7x

Combine like terms: -11 + 11 = 0
7x = 0 + 7x
7x = 7x

Add '-7x' to each side of the equation.
7x + -7x = 7x + -7x

Combine like terms: 7x + -7x = 0
0 = 7x + -7x

Combine like terms: 7x + -7x = 0
0 = 0

Solving
0 = 0

3 0
2 years ago
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