We have,
tan3π24tan5π24tan11π24
=sin3π24sin5π24sin11π24cos3π24cos5π24cos11π24
Multiplying and divide by 2 and we get.
=2sin3π24sin5π24sin11π242cos3π24cos5π24cos11π24
=[cos(3π24−5π24)−cos(3π24+5π24)]sin11π24[cos(3π24+5π24)+cos(3π24−5π24)]cos11π24
=[cosπ12−cosπ3]sin11π24[cosπ3+cosπ12]cos11π24
=[cosπ12−12]sin11π24[12+cosπ12]cos11π24
=[2cosπ12−1]sin11π24[1+2cosπ12]cos11π24
=2cosπ12sin11π24−sin11π24cos11π24+2cosπ12cos11π24
=sin(π12+11π24)−sin(π12−11π24)−sin11π24cos11π24+cos(π12+11π24)+cos(π12−11π24)
=sin(13π24)−sin(−9π24)−sin11π24cos11π24+cos(13π24)+cos(−9π24)
=sin13π24+sin9π24−sin11π24cos11π24+cos13π24+cos9π24
Hence, this is the answer.