Can anyone help me verify: tan 8(x) - tan 8(x)tan^2 4(x) = 2 tan 4 (x) ? I'm stuck on this one and have absolutely no clue as to what to do. Thank you so much for any help! :)

Question

anonymous · Answer

Use the trig identity: tan 2a = 2tan a/(1 - tan^2 a)
Replace a = 4x and 2a = 8x into the identity: tan 8x = (2tan 4x)/(1 - tan^2 4x) (1)
Back to the given equation. Replace tan 8x by the equation (1)
tan 8x - tan 8x(tan^2 4x) = tan 8x (1 - tan^2 4x) = 
= (2tan 4x)* (1 - tan^2 4x)/(1 -tan^2 4x) = 2tan 4x, after simplification.
The equation has been proven.