cosec 48^ + cosec 96^ + cosec 192^ + cosec 384^ =

Step-by-step solution

aligned & cosec 48^ + cosec 96^ + cosec 192^ + cosec 384^ & = cosec (90^ -42^ )+ cosec 96^ + cosec (270^ -78^ ) & + cosec (360^ +24^ ) & = 42^ + cosec 96^ +- 78^ + cosec 24^ & [ cosec (90^ - )= , cosec (270^ - )=- ] & = 1 42^ + 1 96^ - 1 78^ + 1 24^ & = 1 42^ - 1 78^ + 1 96^ + 1 24^ & = ( 78^ - 42^ ) 42^ 78^ + ( 24^ + 96^ ) 96^ 24^ & = -2 60^ 18^ (60^ -18^ ) (60^ +18^ ) & + 2 60^ 36^ (60^ +36^ ) (60^ -36^ ) & = -2 60^ 18^ ^2 (60^ )- ^2 (18^ ) + 2 60^ 36^ ^2 60- ^2 36^ & = -2 3 / 2 ( 5 -1 4 ) (1 / 2)^2- ( 5 -1) 4 )^