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
Artemon [7]
2 years ago
10

If 2tanA=3tanB then prove that,tan(A+B)= 5sin2B/5cos2B-1​

Mathematics
1 answer:
Fed [463]2 years ago
5 0

By definition of tangent,

tan(A + B) = sin(A + B) / cos(A + B)

Using the angle sum identities for sine and cosine,

sin(x + y) = sin(x) cos(y) + cos(x) sin(y)

cos(x + y) = cos(x) cos(y) - sin(x) sin(y)

yields

tan(A + B) = (sin(A) cos(B) + cos(A) sin(B)) / (cos(A) cos(B) - sin(A) sin(B))

Multiplying the right side by 1/(cos(A) cos(B)) uniformly gives

tan(A + B) = (tan(A) + tan(B)) / (1 - tan(A) tan(B))

Since 2 tan(A) = 3 tan(B), it follows that

tan(A + B) = (3/2 tan(B) + tan(B)) / (1 - 3/2 tan²(B))

… = 5 tan(B) / (2 - 3 tan²(B))

Putting everything back in terms of sin and cos gives

tan(A + B) = (5 sin(B)/cos(B)) / (2 - 3 sin²(B)/cos²(B))

Multiplying uniformly by cos²(B) gives

tan(A + B) = 5 sin(B) cos(B) / (2 cos²(B) - 3 sin²(B))

Recall the double angle identities for sin and cos:

sin(2x) = 2 sin(x) cos(x)

cos(2x) = cos²(x) - sin²(x)

and multiplying uniformly by 2, we find that

tan(A + B) = 10 sin(B) cos(B) / (4 cos²(B) - 6 sin²(B))

… = 10 sin(B) cos(B) / (4 (cos²(B) - sin²(B)) - 2 sin²(B))

… = 5 sin(2B) / (4 cos(2B) - 2 sin²(B))

The Pythagorean identity,

cos²(x) + sin²(x) = 1

lets us rewrite the double angle identity for cos as

cos(2x) = 1 - 2 sin²(x)

so it follows that

tan(A + B) = 5 sin(2B) / (4 cos(2B) + 1 - 2 sin²(B) - 1)

… = 5 sin(2B) / (4 cos(2B) + cos(2B) - 1)

… = 5 sin(2B) / (4 cos(2B) - 1)

as required.

You might be interested in
What is the value of 3x^2 + 5x + 25 when x = 3
Leno4ka [110]

Answer: 121

Step-by-step explanation:

6 0
2 years ago
Read 2 more answers
on a map 2 inches equals 40 miles. if two cities are 15 miles apart what is the distinace between them on a map
alexandr402 [8]
On the map it’s 600 inches apart
4 0
2 years ago
Read 2 more answers
Can somebody please help me rn answer <br> ASAP
Viefleur [7K]

Step-by-step explanation:

where is the question huh ?

7 0
2 years ago
keisha made $396 for 18 hours of work at the same rate how many hours would she have to work to make the 242
love history [14]

Answer:

hahaha I stole your points lololoooooooooooollllllllllolololol

Step-by-step explanation:

JK, if you are asking how many hours it would take to make 242 dolla then the answer is 11

4 0
2 years ago
Read 2 more answers
What is 8.478 in expanded form?
nataly862011 [7]
<span>8+0.4+0.07+0.008 you basically separate the numbers starting with the largest working down to the smallest.

i hope i helped

</span>
7 0
3 years ago
Read 2 more answers
Other questions:
  • Veronica got a 5/25 balloon mortgage and her initial payments were $995. She decided to refinance her balloon payment with a 30-
    12·2 answers
  • Earth is approximately 9.3 × 107 miles from the sun. Saturn is approximately 8.87 × 108 miles from the sun. About how much farth
    7·1 answer
  • a concession stand is selling hot dogs and hamburgers during a game. at halftime they sold a total of 78 hot dogs and hamburgers
    11·1 answer
  • I really need help with this question. Can anyone please help me? I have been stuck on this problem for days now.
    8·1 answer
  • What is the area of this green rectangle? | 7 in. 3 in.​
    10·2 answers
  • Solve the equation. x^2=1/49
    10·1 answer
  • 6) A linear equation is graphed. What is the y-intercept of the linear<br> equation? *
    6·1 answer
  • Karma took a 50 question math test, but only 40 questions were correct. What was the percentage of correct answers?
    15·1 answer
  • What is the volume of a cone with base diameter 14 and height 16?
    13·1 answer
  • Mrs. Harrold asked her reading class to make a graph illustrating the similarities and differences between the three stories. Wh
    12·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!