If sin ( alpha + beta) = 4/5, sin ( alpha - beta) =5/13, alpha + beta, alpha - beta, - Maths - Trigonometric Functions

Question

If sin ( alpha + beta) = 4/5, sin ( alpha - beta) =5/13, alpha +
beta, alpha - beta, being acute angles prove that tan 2 alpha =
63/16

Manbar Singh · Accepted Answer

We have,sinα+β = 45⇒cosα+β = 1 - sin2α+β = 1 - 1625 = 35Now, sinα-β = 513⇒cosα-β = 1 - sin2α-β = 1 - 25169 = 1213Now, sin 2α = sinα+β + α-β= sinα+β 
 cosα-β + cosα+β 
 sin α-β=45×1213 + 35×513 = 4865 + 1565 = 6365Now, cos 2α = 1 - sin22α = 1 - 39694225 = 2564225 = 1665Now, tan 2α =  sin 2αcos 2α = 63/6516/65 = 6316

If sin ( alpha + beta) = 4/5, sin ( alpha - beta) =5/13, alpha + beta, alpha - beta, being acute angles prove that tan 2 alpha = 63/16.