<s>[INST] What is the value of this expression: $\log(\tan 6^\circ) + \log (\tan 12^\circ) + \log (\tan 18^\circ)+\cdots+\log(\tan 84^\circ)$ I am not able to solve this problem. Please try it to solve. The log base is 10 and angles are in degree.
Which of the following option is true for the given expression?
(A) Whole number
(B) An irrational number
(C) An negative number
 [/INST] We know $\tan(x^\circ) \tan(90^\circ-x^\circ)=1 \\\tan(x^\circ)=\cot(90^\circ-x^\circ)$
$$6,12,18,\ldots,84\\\text{number of terms}=\frac{\text{last}-\text{first}}{\text{step}}+1=\frac{84-6}{6}+1=13+1=14$$ those are coupled $$6+84=90\\12+78=90\\18+72=90\\\vdots$$so $$\log(\tan 6^\circ) + \log (\tan 12^\circ) + \log (\tan 18^\circ)+\cdots+\log(\tan 84^\circ)=\\
\log(\tan 6^\circ\cdot\tan 12^\circ\cdot\tan 18^\circ\cdots\tan 84^\circ)
=\\\log(\underbrace{\tan 6^\circ\cdot\tan 84^\circ\cdot\tan 12^\circ\cdot\tan 78^\circ\tan 18^\circ\cdot\tan 72^\circ\cdots}_{14 \text{terms}})=\\
\log(\underbrace{1\cdot1\cdot1\cdots}_{7 \text{ terms}})=\\\log(1)=0$$
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