Establish the identity tan 12 theta 3 tan 4 theta tan 34 th

Establish the identity, tan (12 theta) = 3 tan (4 theta)- tan 3(4 theta)/1 -3 tan^2(4 theta) Choose the sequence of steps below that verifies the identity tan (16 theta - 4 theta) = tan (16 theta) + tan (4 theta)/1 - tan (16 theta) tan (4 theta) = 2 tan (8 theta)/1 - tan^2 (8 theta) - tan (4 theta)/1 + 2 tan (8 theta)/1 - tan^2 (8 theta) tan (4 theta) = 3 tan (4 theta) - tan^3 (4 theta)/1 - 3 tan^2 (4 theta) tan (8 theta + 4 theta) = tan (8 theta) - tan (4 theta)/1 + tan (8 theta) tan (4 theta) = 2 tan (4 theta)/1 + tan^2 (4 theta) - tan (4 theta)/1 + 2 than (4 theta)/1 + tan^2 (4 theta) tan (4 theta) = 3 tan (4 theta) - tan^3 (4 theta)/t - 3 tan^2 (4 theta) tan (8 theta + 4 theta) = tan (8 theta) + tan (4 theta)/1 - tan (8 theta) tan (4 theta) = 2 tan (4 theta)/1 - tan^2 (4 theta) + tan (4 theta)/1 - 2 tan (4 theta)/1 - tan^2 (4 theta) tan (4 theta) = 3 tan (4 theta) - tan^3 (4 theta)/1 - 3 tan^2 (4 theta)

Solution

tan(12)

=tan(8+4)...............................................[since tan(A+B)= (tanA+tanB)/(1-tanAtanB)]

=[tan8+tan4]_{/[1-tan8tan4]}

=[[2tan4/(1-tan²4)]+tan4]_{/[1- (2tan4/(1-tan²4))tan4]}

=[[2tan4/(1-tan²4)]+tan4]_{/[1- (2tan²4/(1-tan²4))]}

=[[2tan4+ tan4(1-tan²4)]/(1-tan²4)]_{/[((1-tan²4)- 2tan²4)/(1-tan²4)]}

=[2tan4+ tan4(1-tan²4)]_{/[((1-tan²4)- 2tan²4)]}

=[2tan4+ tan4-tan³4]_{/(1- 3tan²4)}

=[3tan4-tan³4]_{/(1- 3tan²4)}