Question

tan=1/2 tan=1/3 then prove that cos2A=sin4B

Answer 1

Answer:

cos2A = ( 1 - tan ² A ) / ( 1 + tan ² A )

           = ( 1 - 1 / 7² ) / ( 1 + 1 / 7² )

           =  48/49 ×  49/50 = 24/25

cos2A = 24/25

sin4B = sin 2 ( 2B )

          =  [ 2 tan2B / ( 1 + tan² 2B) ]

          = [ 2 { 2 tanB / ( 1 - tan² B ) } ] / [ 1 + { 2 tanB / ( 1 - tan² B ) } ² ]

          = 4 tanB ( 1 - tan² B) / ( 1+ tan⁴B - 2tan²B + 4tan²B )

          = 4 tanB ( 1 - tan² B) / ( 1+ tan⁴B + 2tan²B )

          = 4 tanB ( 1 - tan² B) / ( 1 + tan² B) ²

          = 4 ( 1/3 ) ( 1 - ¹/₉ ) /  ( 1 + ¹/₉ ) ²

          = 24/ 25

sin4B = 24/ 25

Hence proved cos2A = sin4B ...