Establish the identity, tan (12 theta) = 3 tan (4 theta)- tan 3(4 theta)/1 -3 ta

Question

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)

Explanation / Answer

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)

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