Question
Determine the value of \tan20\degree +\tan40\degree+ \sqrt{3}\tan 20\degree\cdotp\tan40\degree
Answer
\tan 60\degree =\tan(20\degree +40\degree )
=\dfrac{\tan20\degree +\tan40\degree}{1-\tan 20\degree\cdotp\tan40\degree} =\sqrt{3}
Therefore,
\tan20\degree +\tan40\degree = \sqrt{3}-\sqrt{3}\tan 20\degree\cdotp\tan40\degree
\tan20\degree +\tan40\degree+ \sqrt{3}\tan 20\degree\cdotp\tan40\degree =\sqrt{3}