Product of tangents -part 2

Using the identity $\tan 3x = \tan (60° - x) \tan x \tan (60° + x)$ ,

If $x = 3°$ , then $\tan 9° = \tan 57° \tan 3° \tan 63°$ $... (A)$

If $x = 9°$ , then $\tan 27° = \tan 51° \tan 9° \tan 69°$ $... (B)$

If $x = 27°$ , then $\tan 81° = \tan 33° \tan 27° \tan 87°$ $... (C)$

Using the identity $\tan 5x = \tan (72° - x) \tan (36° - x) \tan x \tan (36° + x) \tan (72° + x)$ ,

If $x = 3°$ , then $\tan 15° = \tan 69° \tan 33° \tan 3° \tan 39° \tan 75°$ $... (D)$

Equation $(A)$ can be re-written as $\tan 3° = \tan 9° \tan 33° \tan 27°$ , and substituting equation $(B)$ into this gives $\tan 3° = \tan^2 9° \tan 33° \tan 51° \tan 69°$ , which can be re-written as $\tan 3° \tan 39° = \tan^2 9° \tan 33° \tan 69°$ $... (E)$

Equation $(D)$ can be re-written as $\tan^2 15° = \tan 69° \tan 33° \tan 3° \tan 39°$ . Combining this with $(E)$ gives $\tan^2 15° = \tan^2 9° \tan^2 33° \tan^2 69°$ or $\tan 15° = \tan 9° \tan 33° \tan 69°$ $... (F)$

Substituting equation $(F)$ back into $(E)$ gives $\tan 3° \tan 39° = \tan 9° \tan 15°$ , which can be re-written as $\tan 3° \tan 81° \tan 75° = \tan 51°$ , and since equation $(C)$ can be re-written as $\tan 3° \tan 81° = \tan 27° \tan 33°$ , we can substitute that in and obtain $\tan 27° \tan 33° \tan 75° = \tan \boxed{51}°$ .