Question

If sin α=3/5, cos β=12/13 and α, β are positive acute angles, then find the value of (tan α-tan β) / (1+tanαtanβ)

Answer 1

$\huge\boxed{\fcolorbox{green}{pink}{Answer}}$

Construction: 

On the bounding line of the compound angle (α + β) take a point A on OZ, and draw AB and AC perpendiculars to OX and OY respectively. Again, from C draw perpendiculars CD and CE upon OX and AB respectively.

Proof: 

From triangle ACE we get, ∠EAC = 90° - ∠ACE = ∠ECO = alternate ∠COX = α.

Now, from the right-angled triangle AOB we get,

sin (α + β) = ABOAABOA

               = AE+EBOAAE+EBOA

               = AEOAAEOA + EBOAEBOA

               = AEOAAEOA + CDOACDOA

               = AEACAEAC ∙ ACOAACOA + CDOCCDOC ∙ OCOAOCOA

               = cos ∠EAC sin β + sin α cos β

               = sin α cos β + cos α sin β, (since we know, ∠EAC = α)

Therefore, sin (α + β) = sin α cos β + cos α sin β.      (   Proved.)