Hello! Verify the identity. Please show your work! Use trigonometric identities to verify each expression is equal.
Yes this is from edge.
Answers
Answer:
See Below.
Step-by-step explanation:
We want to verify the identity:
[tex]\displaystyle \csc^2 x -2\csc x \cot x +\cot ^2 x = \tan^2\left(\frac{x}{2}\right)[/tex]
Note that the left-hand side is a perfect square trinomial pattern. Namely:
[tex]a^2-2ab+b^2=(a-b)^2[/tex]
If we let a = csc(x) and b = cot(x), we can factor it as such:
[tex]\displaystyle (\csc x - \cot x)^2 = \tan^2\left(\frac{x}{2}\right)[/tex]
Let csc(x) = 1 / sin(x) and cot(x) = cos(x) / sin(x):
[tex]\displaystyle \left(\frac{1}{\sin x}-\frac{\cos x }{\sin x}\right)^2=\tan^2\left(\frac{x}{2}\right)[/tex]
Combine fractions:
[tex]\displaystyle \left(\frac{1-\cos x}{\sin x}\right)^2=\tan^2\left(\frac{x}{2}\right)[/tex]
Square (but do not simplify yet):
[tex]\displaystyle \frac{(1-\cos x)^2}{\sin ^2x}=\tan^2\left(\frac{x}{2}\right)[/tex]
Now, we can make a substitution. Let u = x / 2. So, x = 2u. Substitute:
[tex]\displaystyle \frac{(1-\cos 2u)^2}{\sin ^22u}=\tan^2u[/tex]
Recall that cos(2u) = 1 - sin²(u). Hence:
[tex]\displaystyle \frac{(1-(1-2\sin^2u))^2}{\sin ^2 2u}=\tan^2u[/tex]
Simplify:
[tex]\displaystyle \frac{4\sin^4 u}{\sin ^2 2u}=\tan^2 u[/tex]
Recall that sin(2u) = 2sin(u)cos(u). Hence:
[tex]\displaystyle \frac{4\sin^4 u}{(2\sin u\cos u)^2}=\tan^2 u[/tex]
Square:
[tex]\displaystyle \frac{4\sin^4 u}{4\sin^2 u\cos ^2u}=\tan^2 u[/tex]
Cancel:
[tex]\displaystyle \frac{\sin ^2 u}{\cos ^2 u}=\tan ^2 u[/tex]
Since sin(u) / cos(u) = tan(u):
[tex]\displaystyle \left(\frac{\sin u}{\cos u}\right)^2=\tan^2u=\tan^2u[/tex]
We can substitute u back for x / 2:
[tex]\displaystyle \tan^2\left(\frac{x}{2}\right)= \tan^2\left(\frac{x}{2}\right)[/tex]
Hence proven.
Related Questions
The doll was considered a "feminine" toy. Do these data provide convincing evidence that the mean percentage of the time spent playing with the doll is greater for female monkeys than for male monkeys?
Answers
Answer:
yes it does because men comes before women
Simplify 6V 80.
0 6+16 51
0 6+415
0 6+1615
O 6+4v5i
Answers
Answer:
2nd option
Step-by-step explanation:
Using the rule of radicals
[tex]\sqrt{a}[/tex] × [tex]\sqrt{b}[/tex] ⇔ [tex]\sqrt{ab}[/tex] and [tex]\sqrt{-1}[/tex] = i
Simplifying the radical
[tex]\sqrt{-80}[/tex]
= [tex]\sqrt{16(5)(-1)}[/tex]
= [tex]\sqrt{16}[/tex] × [tex]\sqrt{5}[/tex] × [tex]\sqrt{-1}[/tex]
= 4 × [tex]\sqrt{5}[/tex] × i
= 4i[tex]\sqrt{5}[/tex]
Then
6 + [tex]\sqrt{-80}[/tex]
= 6 + 4i[tex]\sqrt{5}[/tex]
In this addition problem, each letter stands for a different digit.
A
A
+ B B
-------------------
C C C
Find the value of A + B + C?
Answers
Find the circumference of the circle. Round your answer to the nearest
hundredth.
Answers
Answer:
[tex]circumference = 25.13 \ cm[/tex]
Step-by-step explanation:
[tex]circumference = 2 \pi r = 2 \pi 4 = 8 \pi =25.13 \ cm[/tex]
give your answer in detail plzz
Answers
Answer:
60% did not germinate.
Step-by-step explanation:
100% = 5
how many percent are 2 of of these 5 ?
2 = 2/5 × 5 = 2/5 × 100% = 40%
all the other seeds are then the rest of 100%.
100% - 40% = 60%
=>
60% of the seeds did not germinate.
but 40% did.
Find the equation of a line that is parallel to y = x + 1 and passes through the point (3,-4).
Give your answer in the form y = mx + b.
Answers
Given:
The equation of a line is:
[tex]y=x+1[/tex]
A line is parallel to the given line and passes through the point (3,-4).
To find:
The equation of the line in the slope intercept form.
Solution:
The slope intercept form of a line is:
[tex]y=mx+b[/tex] ...(i)
Where, m is the slope and b is the y-intercept of the line.
We have,
[tex]y=x+1[/tex] ...(ii)
On comparing (i) and (ii), we get
[tex]m=1[/tex]
Slope of the given line is 1.
We know that the slopes of parallel lines are equal. So, the slope of the required line is 1.
The required line passes through the point (3,-4) with slope 1. So, the equation of the required line is:
[tex]y-y_1=m(x-x_1)[/tex]
[tex]y-(-4)=1(x-3)[/tex]
[tex]y+4=x-3[/tex]
Subtracting 4 from both sides, we get
[tex]y=x-3-4[/tex]
[tex]y=x-7[/tex]
Therefore, the equation of the required line is [tex]y=x-7[/tex].
if y is inversely proportional to 2x + 1 and the difference in the values of y when x= 0.5 and x= 2 is 0.9, find the value of y when x = -0.25
Answers
Answer:
6
Step-by-step explanation:
answer is in photo above
Vance used the associative property to find the equivalent expression of (4.5 m + StartFraction 7 Over 8 EndFraction) minus 9. His work is shown below.
(4.5 m + StartFraction 7 Over 8 EndFraction) minus 9
Did Vance apply the associative property correctly? Why or why not?
Yes, the terms with variables should be written first.
Yes, the terms can be regrouped and still be equivalent.
No, the order of the terms should have been changed.
No, the addition symbol was dropped in the second expression.
Answers
Answer:
C
Step-by-step explanation:
Answer:
C: Yes, the terms can be regrouped and still be equivalent.
Step-by-step explanation:
Most terms can be regrouped and still be equivalent.
I need help on this one
Answers
Answer:
just look at the constant terms...
A: 5
B: 7
C: -5
1) 11-6 = 5 thus answer A
2) -3 -2 = -5 thus answer C
3) -5 - (12) = -5 + 12 = 7 thus answer B
Step-by-step explanation:
Find the volume of the solid. Round your answer to the nearest hundredth.
Answers
Answer: 27.6
Step-by-step explanation:
tan 36 = 1/x
x =1/ tan 36 x = 1.38 which is your apothem
area of pentagon = 5x2x1.38/2 = 6.9
now do 6.9 x 4 = 27.6
The volume of the solid is 27.52 cubic yards
What is Three dimensional shape?a three dimensional shape can be defined as a solid figure or an object or shape that has three dimensions—length, width, and height.
The given figure is a pentagonal prism
The base length of prism is 2yd
The height of prism is 4 yd
Volume of pentagonal prism V = 1/4 √(5(5+2√5))a²h
a is the base length and h is height
V = 1/4 √(5(5+2√5))4×4
V=27.52
Hence, the volume of the solid is 27.52 cubic yards
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4. Un granjero tiene 57 puercos, que representan el 8 % del total de sus puercos. ¿Cuántos puercos tiene en total?
Answers
Answer:
713 puercos
Step-by-step explanation:
Sea T el número total de puercos
Número de cerdos = 57
Porcentaje = 8%
Para encontrar el número total de puercos;
Porcentaje = 8/100 * T = 57
8T/100 = 57
8T = 5700
T = 5700/8
T = 712,5 ≈ 713 puercos
Por lo tanto, el granjero tiene 713 puercos en total.
If Greg is hosting a BBQ and he needs 50 corn on the cobs. A package of 5 costs $3.25. How much money does he need to get the 50 corn on the cobs?
Answers
Answer: $32.5
Step-by-step explanation:
Given
Greg needs 50 corns on the cobs
A package of 5 costs $3.25
For 50 corns, he needed 10 package
So, cost of 10 packages is
[tex]\Rightarrow 10\times 3.25\\\Rightarrow \$32.5[/tex]
helpp
Plzzz
It’s
A
Question-
Steven earns extra money babysitting. He charges $29.00 for 4 hours and $50.75 for 7
hours.
Enter an equation to represent the relationship. Let x represent the number of hours
Steven babysits and y represent the amount he charges.
Answers
$50.00÷7hours=×$
$50.00÷7hours= 7.25
7.25 he charges per hour
The Jenkins family's monthly budget is
shown in the circle graph. The family has a
monthly income of $4,800. How much
money do they spend on transportation
each month?
Emergency fund
5%
Transportation
5%
Housing
30%
Medical
22%
Food
15%
Clothing
6%
Entertainment
7%
Savings
10%
Answers
Answer:
$240
Step-by-step explanation:
They spend 5% of their monthly income on transportation, so you want to work out 5% of 4800.
An easy trick with 5% is to divide by 10 (drop the ending 0) and then divide by 2. This would be 4800/10/2, or 480/2, which gives 240.
**This content involves using percentages which you may wish to revise.
The equation a + b + c = 180 describes the sum of the angles in a triangle.
Use this formula to find the values of a, b and c when b = 2a and c = 3a
Answers
Answer:
a=30 b=60 c=90
Step-by-step explanation:
you replace b with 2a and c with 3a and you get
a+2a+3a=180
6a=180
a=30
from here you replace a with 30 and find b and c
b=2a=2*30=60
c=3a=3*30=90
Statement ∠T and ∠S are complementary ∠S and ∠U are complementary m∠T+m∠S=90 m∠S+m∠U=90 m∠T+m∠S=m∠S+m∠U m∠U=m∠T ∠U≅∠T
Answers
Answer:
See Explanation
Step-by-step explanation:
According to the Question,
Given That, ∠T and ∠S are complementary & ∠S and ∠U are complementaryThus, m∠T + m∠S = 90 ------ Equation 1
& m∠S + m∠U = 90 ------- Equation 2
Now, Subtract Equation 2 From Equation 1, we getm∠T - m∠U = 0
Thus, m∠T = m∠U
The measure of angle T is equal to angle U.(∠U≅∠T).
Which graph shows a dilation?
Answers
Answer:
A
Step-by-step explanation:
Dialation means to get smaller but it has to stay the same shape and measurements tho.
Find the value of x in the triangle .Please show work
Answers
Answer:
x=4
Step-by-step explanation:
So basically first we see the angles of this triangle, and we learn that all the angles are congruent. Only a equilateral triangle can have all angles congruent, so this must be equilateral triangle. In equilateral triangles all the sides are congruent. So we just put 5x-12=3x-4. We then solve for x and get 4.
What is the area of the shaded sector?
The measure of central angle XYZ is 1.257 radians.
O 101 units
O 207 units
O 40 units
O 80 units
N
8
X
Answers
The area of the sector of the given circle is: 40 units²
What is circle?A circle is a round-shaped figure that has no corners or edges. In geometry, a circle can be defined as a closed, two-dimensional curved shape.
The set of all points on a plane that are a fixed distance from a center,
A circle is a special kind of ellipse in which the eccentricity is zero and the two foci are coincident.
The circle is the shape with the largest area for a given length of perimeter.
Given the following:
r = 8
θ = 1.25 radians
Area of sector = (1/2) × r²θ = (1/2) × (8²)(1.25)
Hence, the Area of sector = 40 units²
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What is the surface area of a cube that measures 4 inches on each side?
____ square inches
Answers
Answer:
96 Square inches
Step-by-step explanation:
because the formula of the surface of the cube is l x l x 6 so 4 x 4 x 6
excuse me for my bad English but I am italian
(a+b)(a-b)=? Đây là hằng đẳng thức nào?
Answers
Answer:
không
Step-by-step explanation:
Answer:
(a + b)(a - b) = a² - b²
Step-by-step explanation:
(a + b)(a - b) = a (a - b ) + b (a - b )
= a² - ab + ba - b² [ ab = ba ]
= a² - ab + ab - b² [ ab - ab = 0 ]
= a² - b²
how many ways can a person toss a coin 14 times so that tge number of tails is between 7 and 12 inclusive
Answers
Answer:
8 tails I think so If the answer is correct plz mark me the brainliest
Neil is buying steak for a cookout on Saturday. Steak is on sale for $9.62 per pound. If he buys 7.5 pounds of steak, how much money does he spend?
Answers
Answer:
$72.15
Step-by-step explanation:
x= pounds of steak
$[tex]9.62x[/tex] = money spent on steaks
If Neil buys 7.5 pounds of steak then x =7.5
$9.62(7.5) = $72.15
Answer:
$72.15
Step-by-step explanation:
1 pound = $9.62
7.5 pounds = $9.62 × 7.5
= $72.15
find the volume of this cone. round to the nearest tenth. l=10 r-6
Answers
Answer:
376.8
Step-by-step explanation:
Formula Volume Cone = πr² * h/3
Volume = 3.14 * 36 * 3 1/3
Volume ≈ 376.8
If my answer is incorrect, pls correct me!
If you like my answer and explanation, mark me as brainliest!
-Chetan K
Answer: 301.6
-------------------
answer above isn't right this is, trust me.
The function h is defined by h (x) = 3x2 + 7.
Find h(5x).
h (5x) = ?
Answers
Answer:
h(5x) = 75x² + 7
Step-by-step explanation:
h(x) = 3x² + 7
h(5x) = 3(5x)² + 7
h(5x) = 3(25x²) + 7
h(5x) = 75x² + 7
[tex]h(5x) = 75x^2 + 7[/tex]
To find h(5x), you simply need to substitute 5x in place of x in the function [tex]h(x) = 3x^2 + 7[/tex] and then simplify the expression.
[tex]h(x) = 3x^2 + 7[/tex]
Now, replace x with [tex]5x: h(5x) = 3(5x)^2 + 7[/tex]
Simplify: [tex]h(5x) = 3(25x^2) + 7[/tex]
Now, calculate [tex]3 * 25: h(5x) = 75x^2 + 7[/tex]
So, [tex]h(5x) = 75x^2 + 7[/tex]
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During a nature walk, Mary identified
24 species of animals and plants.
Mary said
1/5
of the species she identified were
animals. Can this be correct? Explain.
Answers
Answer:
No, it cannot be correct.
Step-by-step explanation:
1/5 of 24 is 4.8.
This would mean that Mary would have identified 4.8 animals.
This is logically incorrect, because the number of animals has to be a full number. There cannot be any decimals, or parts of animals.
So, this cannot be correct.
If 5 + 2 root 3 / 7 + 4 root 3 = a - b root 3, find a and b.
Answers
Answer:
lhs = (5+2√3)/(7+4√3)
= [(5+2√3)(7-4√3)]/[(7-4√3)(7+4√3)]
=[35-20√3+14√3-24]/[7²-(4√3)²]
=[11-6√3]/[49-48]
=11-6√3
therefore
11-6√3 = a+b√3
compare both sides
a= 11, b= -6
Step-by-step explanation:
Hope this answer helps you :)
Have a great day
Mark brainliest
Answer:
a = 11, b = 6
Step-by-step explanation:
Given
[tex]\frac{5+2\sqrt{3} }{7+4\sqrt{3} }[/tex]
Rationalise the denominator by multiplying numerator/ denominator by the conjugate of the denominator.
The conjugate of 7 + 4[tex]\sqrt{3}[/tex] is 7 - 4[tex]\sqrt{3}[/tex]
= [tex]\frac{(5+2\sqrt{3})(7-4\sqrt{3}) }{(7+4\sqrt{3})(7-4\sqrt{3} }[/tex] ← expand numerator/ denominator using FOIL
= [tex]\frac{35-20\sqrt{3}+14\sqrt{3}-24 }{49-28\sqrt{3}+28\sqrt{3}-48 }[/tex]
= [tex]\frac{11-6\sqrt{3} }{1}[/tex]
= 11 - 6[tex]\sqrt{3}[/tex]
with a = 11 and b = 6
As an enrichment treat, your school has been given 63 tickets to go to a local theme park for FREE! You need to work out who is getting the tickets. The Principal decides that each of the eight classes in your school will get the same number of tickets. How many tickets will each class get and how many will be left over?
Tickets per class answer:
Remaining tickets answer:
please explain
Answers
63 / 8 = 7 remainder 7
7 x 8 = 56 so there are 7 tickets remaining
Which is the Value if this expression when m = 3 and n = -5
Answers
A parabola has a vertex at (–5, 0) and passes through the point (–3, 1). Which of the following is the equation of this parabola in the form y=a(x-h)^2+k?
Answers
Answer:
y = 1/4(x + 5)^2
Step-by-step explanation:
Using the given equation form;
(h,k) represents the vertex of the parablola
To get the value of a, we use the given point
So we have;
1 = a(-3 + 5)^2 + 0
1 = a(2)^2
4a = 1
a = 1/4
so we have the equation as;
y = 1/4(x+5)^2 + 0
y = 1/4(x + 5)^2
