MATHEMATICS COLLEGE

Prove the trigonometric identity
(tan x + cot x)/(csc x * cos x) = sec^2 x

Answers

Answer 1

Answer:

Answer:

$(\tan x + \cot x)/(\csc x \cos x)=\sec^2 x$

$\boxed{((\sin x)/(\cos x) + (\cos x)/(\sin x))/((1)/(\sin x) \cdot \cos x)}=\sec^2 x$

$\boxed{((\sin^2 x)/(\sin x\cos x) + (\cos^2 x)/(\sin x \cos x))/((\cos x)/(\sin x))}=\sec^2 x$

$\boxed{((\sin^2 x+\cos^2 x)/(\sin x\cos x))/((\cos x)/(\sin x))}=\sec^2 x$

$\boxed{((1)/(\sin x\cos x))/((\cos x)/(\sin x))}=\sec^2 x$

$\boxed{(1)/(\sin x\cos x) \cdot (\sin x)/(\cos x)}=\sec^2 x$

$(1)/(\cos^2x)=\sec^2x$

$\sec^2x=\sec^2x$

Step-by-step explanation:

Given trigonometric identity:

$(\tan x + \cot x)/(\csc x \cos x)=\sec^2 x$

$\textsf{Use the identities\;\;$\tan x = (\sin x)/(\cos x)$\;,\;$\cot x=(\cos x)/(\sin x)$\;\;and\;\;$\csc x=(1)/(\sin x)$}:$

$\boxed{((\sin x)/(\cos x) + (\cos x)/(\sin x))/((1)/(\sin x) \cdot \cos x)}=\sec^2 x$

Simplify the denominator and make the fractions in the numerator like fractions:

$\boxed{((\sin^2 x)/(\sin x\cos x) + (\cos^2 x)/(\sin x \cos x))/((\cos x)/(\sin x))}=\sec^2 x$

$\textsf{Apply\;the\;fraction\;rule\;\;$(a)/(b)+(c)/(b)=(a+c)/(b)$\;to\;the\;numerator}:$

$\boxed{((\sin^2 x+\cos^2 x)/(\sin x\cos x))/((\cos x)/(\sin x))}=\sec^2 x$

$\textsf{Use\;the\;identity\;\;$\sin^2x+\cos^2x=1$}:$

$\boxed{((1)/(\sin x\cos x))/((\cos x)/(\sin x))}=\sec^2 x$

$\textsf{Apply\;the\;fraction\;rule\;\;$(a)/((b)/(c))=a \cdot (c)/(b)$}:$

$\boxed{(1)/(\sin x\cos x) \cdot (\sin x)/(\cos x)}=\sec^2 x$

Cancel the common factor sin x, and apply the exponent rule aa = a² to the denominator:

$(1)/(\cos^2x)=\sec^2x$

$\textsf{Use the identity\;\;$(1)/(\cos x)=\sec x$}:$

$\sec^2x=\sec^2x$

Answer 2

Answer:

Answer:

The proof of the trigonometric identity:

We can start by expanding the numerator and denominator. In the numerator, we can use the trigonometric identities tan x = sin x / cos x and cot x = cos x / sin x.

In the denominator, we can use the trigonometric identity csc x = 1 / sin x. This gives us:

$((tan x + cot x))/((csc x * cos x) ) = (((sin x )/( cos x)) + ((cos x )/(sin x)))/(((1)/( sin x)) * cos x)$

`We can then cancel the sin x terms in the numerator and denominator. This gives us:

$((tan x + cot x))/((csc x * cos x) ) = (1 + 1)/(((1 )/(sin x)) * cos x)$

We can then multiply the numerator and denominator by sin x. This gives us:

$((tan x + cot x))/((csc x * cos x) ) = (sin x + sin x)/((1 )/(cos x))$

We can then simplify the expression. This gives us:

$((tan x + cot x))/((csc x * cos x) ) = (2sin x)/((1 )/(cos x)) = (2sin x)/(cos x) = 2tan x$

Finally, we can use the trigonometric identity tan^2 x = sec^2 x - 1 to get:

$2tan x =( 2tan^2 x )/( (sec^2 x - 1))$

This gives us the following identity:

$((tan x + cot x))/((csc x * cos x) ) = sec^2 x$

This completes the proof of the trigonometric identity.

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QuestionEnter the exponential function using t (for time) as the independent variable to model the situation. Thenfind the value of the function after the given amount of time.A new savings account is opened with $400 and gains 3.5% yearly for 5 years.The exponential function that models the situation is y = MyAfter 5 years, the savings account has $

Answers

Since, it is an exponential function, thus this is a compound interest problem;

Where; the function is given as;

$\begin{gathered} A(t)=P(1+r)^t \n \text{Where A(t)= amount in the savings account at a time t} \n P=ca\text{ pital invested} \n r=\text{rate } \n t=\text{ time} \end{gathered}$

Thus, the function required is;

$\begin{gathered} A(t)=400(1+(3.5)/(100))^t \n A(t)=400(1.035)^t \end{gathered}$ $\begin{gathered} y=400(1.035)^t \n \text{Where t is the time} \end{gathered}$

After 5 years,

$\begin{gathered} A(5)=400(1.035)^5 \n A(5)=400(1.1877) \n A(5)=475.07 \end{gathered}$

The amount in the savings account after five years is $475.07

-5+i/2i how do I break this down?

Answers

Answer:

i2 = -1

Step-by-step explanation:

5i ⋅i⋅(−2i)= −

10 ⋅ i2 ⋅ i= − 10 ⋅ (−1) ⋅ i = 10i

Renting a car costs $30 per day, or $600 per month. Renting daily is cheaper for a few days, but after how many days are the two options equal (after which renting monthly is cheaper)?

Answers

Answer:

after 20 days it will be equal

Step-by-step explanation:

i dont have step by step but the problome is 30x20=600

PLEASE ANSWER ASAP!!! FULL ANSWERS ONLY!!!!!! WILL GIVE BRAINLIEST!!!!!!!!! Two cyclists, 68 miles apart, start riding toward each other at the same time. One cycles 3 miles per hour faster than the other, and they meet after 4 hours of riding.

a. Write an equation using the information as it is given above that can be solved to answer this problem. Use the variable r to represent the speed of the slower cyclist.

b. What are the speeds of the two cyclists? _______________

Answers

a) The given equation using the information as it is given above that can be solved to answer this problem is 4r + 4(r+3) = 68

b) The speed of both cyclists are 3mi/hr and 10mi/hr respectively

The formula for calculating the distance covered is expressed as:

Distance = speed * time

Let the speed covered by one cyclist be t

If one cycles 3 miles per hour faster than the other, the speed of the other will be t + 3

If both meet after 4 hours of riding, then their time will be 4 hours

Distance covered the first cyclist = 4r

Distance covered by the second = 4(r+3)

If the two cyclists are 68 miles apart, then:

4r + 4(r+3) = 68

a) The given equation using the information as it is given above that can be solved to answer this problem is 4r + 4(r+3) = 68

b) Expand the equation in a to get "r"

4r + 4r + 12 = 68

8r + 12 = 68

8r = 68 - 12

8r = 56

r = 56/8

r = 7miles per hour

Speed of the first cyclist = 7mi/hr

Speed of the second cyclist = 3 + 7mi/hr = 10mi/hr

Hence the speed of both cyclists are 3mi/hr and 10mi/hr respectively

Learn more here: brainly.com/question/19915685

Answer:

4r + 4(r + 3) = 68

r = 7 miles per hour

r + 3 = 10 miles per hour

Step-by-step explanation:

distance = rate * time

4r + 4(r + 3) = 68

Distribute

4r + 4r + 12 = 68

8r + 12 = 68

8r = 56

r = 7 miles per hour

r + 3 = 10 miles per hour

Whats 1/10 of 6,000

Answers

Answer:

600

Step-by-step explanation:

(1 / 10) * 6000

That's it!

PLEASE HELP QUICK I NEED IT TO PASS. PLEASE SHOW STEPS, AND DON'T SPAM LINK PLEASE HELP. I NEED SOME HELP

Answers

Answer: 151

Step-by-step

you need to first find the area of each piece of the tent

the 2 rectangular flaps would be 6*8= 48 each

the bottom would be 4*8=32

to find the area of the triangular pieces we need to know the height of each triangle, right now we only know sides. To find the height (the altitude) you draw a line straight down to make a 90 degree angle.

that will also cut your base in half (2 instead of 4) and make 6 your hypotenuse

now use the Pythagorean theorem to find the height of the triangle

$a^(2) +b^(2) =c^(2) \n\na^(2) +2^(2) =6^(2) \na^(2) +4=36\na^(2) =32\na=√(32) = 5.66$