1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Alina [70]
3 years ago
6

Please someone help me to prove this. ​

Mathematics
2 answers:
liubo4ka [24]3 years ago
6 0

Answer:  see proof below

<u>Step-by-step explanation:</u>

Use the Power Reducing Identity:  sin² Ф = (1 - cos 2Ф)/2

Use the Double Angle Identity:  sin 2Ф = 2 sin Ф · cos Ф

Use the following Sum to Product Identities:

\sin x - \sin y = 2\cos \bigg(\dfrac{x+y}{2}\bigg)\sin \bigg(\dfrac{x-y}{2}\bigg)\\\\\\\cos x - \cos y = -2\sin \bigg(\dfrac{x+y}{2}\bigg)\sin \bigg(\dfrac{x-y}{2}\bigg)

<u>Proof LHS →  RHS</u>

\text{LHS:}\qquad \qquad \qquad \dfrac{\sin^2A-\sin^2B}{\sin A\cos A-\sin B \cos B}

\text{Power Reducing:}\qquad \dfrac{\bigg(\dfrac{1-\cos 2A}{2}\bigg)-\bigg(\dfrac{1-\cos 2B}{2}\bigg)}{\sin A \cos A-\sin B\cos B}

\text{Half-Angle:}\qquad \qquad \dfrac{\bigg(\dfrac{1-\cos 2A}{2}\bigg)-\bigg(\dfrac{1-\cos 2B}{2}\bigg)}{\dfrac{1}{2}\bigg(\sin 2A-\sin 2B\bigg)}

\text{Simplify:}\qquad \qquad \dfrac{1-\cos 2A-1+\cos 2B}{\sin 2A-\sin 2B}\\\\\\.\qquad \qquad \qquad =\dfrac{-\cos 2A+\cos 2B}{\sin 2A - \sin 2B}\\\\\\.\qquad \qquad \qquad =\dfrac{\cos 2B-\cos 2A}{\sin 2A-\sin 2B}

\text{Sum to Product:}\qquad \qquad \dfrac{-2\sin \bigg(\dfrac{2B+2A}{2}\bigg)\sin \bigg(\dfrac{2B-2A}{2}\bigg)}{2\cos \bigg(\dfrac{2A+2B}{2}\bigg)\sin \bigg(\dfrac{2A-2B}{2}\bigg)}

\text{Simplify:}\qquad \qquad \dfrac{-2\sin (A + B)\cdot \sin (-[A - B])}{2\cos (A + B) \cdot \sin (A - B)}

\text{Co-function:}\qquad \qquad \dfrac{2\sin (A + B)\cdot \sin (A - B)}{2\cos (A + B) \cdot \sin (A - B)}

\text{Simplify:}\qquad \qquad \quad \dfrac{\cos (A+B)}{\sin (A+B)}\\\\\\.\qquad \qquad \qquad \quad =\tan (A+B)

LHS = RHS:    tan (A + B) = tan (A + B)    \checkmark

dem82 [27]3 years ago
5 0
<h3><u>Answer</u> :</h3>

We know that,

\dag\bf\:sin^2A=\dfrac{1-cos2A}{2}

\dag\bf\:sin2A=2sinA\:cosA

<u>_</u><u>_</u><u>_</u><u>_</u><u>_</u><u>_</u><u>_</u><u>_</u><u>_</u><u>_</u><u>_</u><u>_</u><u>_</u><u>_</u><u>_</u><u>_</u><u>_</u><u>_</u><u>_</u><u>_</u><u>_</u><u>_</u><u>_</u><u>_</u><u>_</u><u>_</u><u>_</u><u>_</u><u>_</u><u>_</u><u>_</u><u>_</u><u>_</u><u>_</u><u>_</u>

<u>Now, Let's solve</u> !

\leadsto\:\bf\dfrac{sin^2A-sin^2B}{sinA\:cosA-sinB\:cosB}

\leadsto\:\sf\dfrac{\frac{1-cos2A}{2}-\frac{1-cos2B}{2}}{\frac{2sinA\:cosA}{2}-\frac{2sinB\:cosB}{2}}

\leadsto\:\sf\dfrac{1-cos2A-1+cos2B}{sin2A-sin2B}

\leadsto\:\sf\dfrac{2sin\frac{2A+2B}{2}\:sin\frac{2A-2B}{2}}{2sin\frac{2A-2B}{2}\:cos\frac{2A+2B}{2}}

\leadsto\:\sf\dfrac{sin(A+B)}{cos(A+B)}

\leadsto\:\bf{tan(A+B)}

You might be interested in
The graph represents function 1, and the equation represents function 2:
Veseljchak [2.6K]

If there was a graphical representation, I would be happy to assist you.

6 0
3 years ago
Find the distance between the two given points H(-3,6) and C(-3,-3)
sweet [91]

Answer:

Similarly, the distance between two points P1 = (x1,y1,z1) and P2 = (x2,y2,z2) in xyz-space is given by the following generalization of the distance formula, d(P1,P2) = (x2 x1)2 + (y2 y1)2 + (z2 z1)2. This can be proved by repeated application of the Pythagorean Theorem.

Step-by-step explanation:

Use a calculator  

3 0
3 years ago
Find the circumference and area of a circle with diameter 2 ft
iren2701 [21]

Answer:

C = 6.28

A = 3.14

Step-by-step explanation:

use the equations for area and circumference to solve

C = pi d

A = pi r^2

I like to remember these using the saying Cherry pie's delicious, Apple pie's are too so Cherry = pi's delicious, Apple = pi's r 2

so with the diameter of 2 plug in to get circumference

C = 3.14 (2)

C = 6.28

and the equation for area uses radius not diameter but we know that d/2=r so radius is 1, plug in to solve

A = 3.14 (1)^2

A = 3.14 (1)

A = 3.14

Hope this helps! :)

6 0
2 years ago
Alan is going rollerblading it cost $2.75 to rent rollerblades and $1.50 for each hour spent rollerblading if alex needs to rent
ella [17]
8.75 I think bc 1.50 x 4 = 6 then add 2.75 and you get $8.75
3 0
3 years ago
Could u all check this if am correct before i turn it in.​
just olya [345]
All the answers are correct.
8 0
2 years ago
Read 2 more answers
Other questions:
  • Ms.peterson class sold tickets for the class play.how many tickets in all did brandon and haylie sell
    11·1 answer
  • HELP PLEASE???!!!!!!!
    9·1 answer
  • Point) a norman window has the shape of a rectangle with a semi circle on top; diameter of the semicircle exactly matches the wi
    14·1 answer
  • What is 3x3 – 11x2 – 26x + 30 divided by x – 5?
    14·1 answer
  • Can someone help me with this please i’ll give brainliest and it’s due in a half hour
    11·1 answer
  • How many megabyte equals 32gigabyte​
    5·2 answers
  • According to the Bureau of the Census, the population of the United States in 1990 was approximately 249 million and in 2000, 28
    14·1 answer
  • Help me pleaseeeeeeeeeeeeeeeee
    7·1 answer
  • Help! Give brainliest!
    10·2 answers
  • Arrange the given the decimals in ascending order from least to greatest ​
    9·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!