Question

prove that tan12 tan24 tan48 tan84=1

Answer 1

if you remember 
tan x tan (60-x) tan (60+x) = tan 3x as per http://math-kali.blogspot.in/2015/08/201... 

put x = 12 to get 
tan 12 tan 48 tan 72 = tan 36 ...(1) 
put x = 24 to get 
tan 24 tan 36 tan 84 = tan 72 .. (2) 
multiply (1) and (2) to get the result.

Answer 2

Answer:

To prove: tan 12 . tan 24 . tan 48 . tan 84 = 1

we use the following result to Prove it,

tan x . tan (60-x)  . tan (60+x) = tan 3x  

for given statement,

put x = 12

we get

tan 12 . tan 48 . tan 72 = tan 36 ........................(1) 

now, put x = 24

we get 

tan 24 . tan 36 . tan 84 = tan 72 .......................(2) 

Multiply (1) and (2), we get

tan 12 . tan 48 . tan 72 . tan 24 . tan 36 . tan 84 = tan 36 . tan 72

transpose tan 36 & tan 72 to RHS, we get

$tan\,12\,.\,tan\,24\,.\,tan\,48\,.\,tan\,84=\frac{tan\,36\,.\,tan\,72}{tan\,36\,.\,tan\,72}$

$tan\,12\,.\,tan\,24\,.\,tan\,48\,.\,tan\,84=1$

Hence Proved