JEE Exam > JEE Questions > If (sin alpha)/(sin beta) (cos alpha)/(cos be...
Start Learning for Free
If (sin alpha)/(sin beta) (cos alpha)/(cos beta) 1=0(alpha beta!=(n pi)/(2) n in z) then (sin^(3)beta)/(sin alpha) (cos^(3)beta)/(cos alpha)=?
Most Upvoted Answer
If (sin alpha)/(sin beta) (cos alpha)/(cos beta) 1=0(alpha beta!=(n pi...
Solution:
Given, (sin α/sin β) (cos α/cos β) = 1 and αβ ≠ (nπ/2) where n ∈ ℤ
We need to find (sin³β/sin α) (cos³β/cos α)
Let's simplify the given expression first.
(sin α/sin β) (cos α/cos β) = 1
Taking LCM, we get
(cos α sin α)/(cos β sin β) = 1
Cross-multiplying, we get
cos α sin α = cos β sin β
Dividing by cos α sin β, we get
cot β = cot α
Since αβ ≠ (nπ/2), we can say that α ≠ β.
Therefore, cot β = cot α implies β = α + nπ where n ∈ ℤ.
Now, let's simplify the expression we need to find.
(sin³β/sin α) (cos³β/cos α)
= (sin²β cos²β/sin α cos α) (sin β cos β)
= [(1-cos²β)(cos²β)/sin α cos α] (sin β cos β)
= [(cos²β-sin²β cos²β)/sin α cos α] (sin β cos β)
= [(cos²β(1-sin²β))/sin α cos α] (sin β cos β)
= [(cos²β cos²α)/sin α cos α] (sin β cos β) (using cot β = cot α)
= cos²β sin β cos β
= (cos³β sin β)/cos β
= cos²β sin β
= sin β cos²β
Therefore, (sin³β/sin α) (cos³β/cos α) = sin β cos²β
Hence, the required expression is sin β cos²β.
Given, (sin α/sin β) (cos α/cos β) = 1 and αβ ≠ (nπ/2) where n ∈ ℤ
We need to find (sin³β/sin α) (cos³β/cos α)
Let's simplify the given expression first.
(sin α/sin β) (cos α/cos β) = 1
Taking LCM, we get
(cos α sin α)/(cos β sin β) = 1
Cross-multiplying, we get
cos α sin α = cos β sin β
Dividing by cos α sin β, we get
cot β = cot α
Since αβ ≠ (nπ/2), we can say that α ≠ β.
Therefore, cot β = cot α implies β = α + nπ where n ∈ ℤ.
Now, let's simplify the expression we need to find.
(sin³β/sin α) (cos³β/cos α)
= (sin²β cos²β/sin α cos α) (sin β cos β)
= [(1-cos²β)(cos²β)/sin α cos α] (sin β cos β)
= [(cos²β-sin²β cos²β)/sin α cos α] (sin β cos β)
= [(cos²β(1-sin²β))/sin α cos α] (sin β cos β)
= [(cos²β cos²α)/sin α cos α] (sin β cos β) (using cot β = cot α)
= cos²β sin β cos β
= (cos³β sin β)/cos β
= cos²β sin β
= sin β cos²β
Therefore, (sin³β/sin α) (cos³β/cos α) = sin β cos²β
Hence, the required expression is sin β cos²β.
Attention JEE Students!
To make sure you are not studying endlessly, EduRev has designed JEE study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in JEE.
|
Explore Courses for JEE exam
|
|
Similar JEE Doubts
Top Courses for JEEView all
If (sin alpha)/(sin beta) (cos alpha)/(cos beta) 1=0(alpha beta!=(n pi)/(2) n in z) then (sin^(3)beta)/(sin alpha) (cos^(3)beta)/(cos alpha)=?
Question Description
If (sin alpha)/(sin beta) (cos alpha)/(cos beta) 1=0(alpha beta!=(n pi)/(2) n in z) then (sin^(3)beta)/(sin alpha) (cos^(3)beta)/(cos alpha)=? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If (sin alpha)/(sin beta) (cos alpha)/(cos beta) 1=0(alpha beta!=(n pi)/(2) n in z) then (sin^(3)beta)/(sin alpha) (cos^(3)beta)/(cos alpha)=? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If (sin alpha)/(sin beta) (cos alpha)/(cos beta) 1=0(alpha beta!=(n pi)/(2) n in z) then (sin^(3)beta)/(sin alpha) (cos^(3)beta)/(cos alpha)=?.
If (sin alpha)/(sin beta) (cos alpha)/(cos beta) 1=0(alpha beta!=(n pi)/(2) n in z) then (sin^(3)beta)/(sin alpha) (cos^(3)beta)/(cos alpha)=? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If (sin alpha)/(sin beta) (cos alpha)/(cos beta) 1=0(alpha beta!=(n pi)/(2) n in z) then (sin^(3)beta)/(sin alpha) (cos^(3)beta)/(cos alpha)=? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If (sin alpha)/(sin beta) (cos alpha)/(cos beta) 1=0(alpha beta!=(n pi)/(2) n in z) then (sin^(3)beta)/(sin alpha) (cos^(3)beta)/(cos alpha)=?.
Solutions for If (sin alpha)/(sin beta) (cos alpha)/(cos beta) 1=0(alpha beta!=(n pi)/(2) n in z) then (sin^(3)beta)/(sin alpha) (cos^(3)beta)/(cos alpha)=? in English & in Hindi are available as part of our courses for JEE.
Download more important topics, notes, lectures and mock test series for JEE Exam by signing up for free.
Here you can find the meaning of If (sin alpha)/(sin beta) (cos alpha)/(cos beta) 1=0(alpha beta!=(n pi)/(2) n in z) then (sin^(3)beta)/(sin alpha) (cos^(3)beta)/(cos alpha)=? defined & explained in the simplest way possible. Besides giving the explanation of
If (sin alpha)/(sin beta) (cos alpha)/(cos beta) 1=0(alpha beta!=(n pi)/(2) n in z) then (sin^(3)beta)/(sin alpha) (cos^(3)beta)/(cos alpha)=?, a detailed solution for If (sin alpha)/(sin beta) (cos alpha)/(cos beta) 1=0(alpha beta!=(n pi)/(2) n in z) then (sin^(3)beta)/(sin alpha) (cos^(3)beta)/(cos alpha)=? has been provided alongside types of If (sin alpha)/(sin beta) (cos alpha)/(cos beta) 1=0(alpha beta!=(n pi)/(2) n in z) then (sin^(3)beta)/(sin alpha) (cos^(3)beta)/(cos alpha)=? theory, EduRev gives you an
ample number of questions to practice If (sin alpha)/(sin beta) (cos alpha)/(cos beta) 1=0(alpha beta!=(n pi)/(2) n in z) then (sin^(3)beta)/(sin alpha) (cos^(3)beta)/(cos alpha)=? tests, examples and also practice JEE tests.
|
|
Explore Courses for JEE exam
|
|
Suggested Free Tests
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
Follow Us
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup