If sin x ÷a= cos x ÷b , prove that a sin 2x + b cos 2x = b

tan x = a/b

cos x = b/ (a2+b2)1/2 ,  , sinx = a/ (a2+b2)1/2

 b cos2x + a sin 2x = b (2cos2x-1) + a (2 sin x . cosx)

 b( 2 .b2/a2+b2-1) + a ( 2 .a/(a2+b2)1/2.b/(a2+b2)1/2

= b(2b2- a2-b2)/a2+b2 + (2a2b/ a2+b2)

 =b (b2-a2)/a2+b2 +2a2b/a2+b2

 =b (b2-a2+2a2)/a2+b2

= b = R.H.S