Trigonometric question

Question

If tan x = 3 tan y, prove that tan(x − y) = (sin 2y) / (2−cos2y), You can only work on one side of the identity.

Answer 1

Given: tan(x) = 3*tan(y)

Prove: tan(x - y) = sin(2y) / [ 2 - cos(2y) ]

PROOF:

sin(2y) / [ 2 - cos(2y) ] = tan(x - y) Given

= [ tan(x) - tan(y) ] / [ 1 + tan(x)*tan(y) ] Difference Formula for tan(x-y)

= [ 3*tan(y) - tan(y)] / [ 1 + 3*tan(y)*tan(y) ] Substutite for tan(x)

= [ 2*tan(y)] / [ 1 + 3*[tan(y)]^2 ] Now, RHS in terms of y only

= [ 2*tan(y)] / { [ (cos(y))^2 + 3*(sin(y))^2 ] / (cos(y))^2 }

= [ 2*tan(y)* (cos(y))^2 ] / [ (cos(y))^2 + 3*(sin(y))^2 ]

= [ 2*tan(y)*(cos(y))^2 ] / [ (cos(y))^2 + 3*(sin(y))^2 ]

= [ 2*tan(y)*(cos(y))^2 ] / [ 1 + 2*(sin(y))^2 ]

= [ 2*sin(y)*cos(y) ] / [ 1 + 2*( (1 + cos(2y)) / 2 ) ]

= [ sin(2y) ] / [ 1 + 1 + cos(2y) ]

= [ sin(2y) ] / [ 2 + cos(2y) ]

= [ sin(2y) ] / [ 2 - cos(2y) ] Because cos(u) = - cos(u)

DONE!