Prove the following: If sin 2A = λsin 2B then prove that tan(A+B)tan(A-B)=λ+1λ-1 - Mathematics and Statistics

Sum

Prove the following:

If sin 2A = λsin 2B then prove that (tan("A" + "B"))/(tan("A" - "B")) = (lambda + 1)/(lambda - 1)

Solution

sin 2A = λsin 2B

∴ (sin 2"A")/(sin 2"B") = lambda/1

∴ (sin2"A" + sin2"B")/(sin2"A" - sin2"B") = (lambda + 1)/(lambda - 1)

∴ (2sin((2"A" + 2"B")/2)*cos((2"A" - 2"B")/2))/(2cos ((2"A" + 2"B")/2)*sin((2"A" - 2"B")/2)) = (lambda + 1)/(lambda - 1)

∴ (sin("A" + "B")*cos("A" - "B"))/(cos("A" + "B")*sin("A" - "B")) = (lambda + 1)/(lambda - 1)

∴ tan(A + B) · cot(A – B) = (lambda + 1)/(lambda - 1)

∴ (tan("A" + "B"))/(tan("A" - "B")) = (lambda + 1)/(lambda - 1).

Concept: Trigonometric Functions of Sum and Difference of Angles
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Balbharati Mathematics and Statistics 1 (Arts and Science) 11th Standard Maharashtra State Board
Chapter 3 Trigonometry - 2
Miscellaneous Exercise 3 | Q II. (12) | Page 57