Question: The value of $ \cot 41{}^\circ \cdot \cot 42{}^\circ \cdot \cot 43{}^\circ \cdot \cot 44{}^\circ $ $ \cdot \cot 45{}^\circ \cdot \cot 46\cdot \cot 47{}^\circ \cdot \cot 48{}^\circ \cdot cot49{}^\circ $ is
Options:
A) 0
B) 1
C) $ \frac{1}{\sqrt{2}} $
D) $ \frac{\sqrt{3}}{2} $
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Answer:
Correct Answer: B
Solution:
- [b] $ \cot 41{}^\circ \cdot \cot 42{}^\circ \cdot \cot 43{}^\circ \cdot \cot 44{}^\circ $
$ \cdot \cot 45{}^\circ \cdot \cot 46{}^\circ \cdot \cot 47{}^\circ \cdot \cot 48{}^\circ \cdot \cot 49{}^\circ $
$ =\cot (90{}^\circ -49{}^\circ )\cdot \cot (90{}^\circ -48{}^\circ )\cdot \cot (90{}^\circ -47{}^\circ ) $
$ \cdot \cot (90{}^\circ -46{}^\circ )\cdot \cot 45{}^\circ -\cot 46{}^\circ $
$ \cdot \cot 47{}^\circ \cdot \cot 48{}^\circ \cdot \cot 49{}^\circ $
$ =\tan 49{}^\circ \cdot \tan 48{}^\circ \cdot \tan 47{}^\circ \cdot \tan 46{}^\circ \cdot \cot 45{}^\circ $
$ \cdot \cot 46{}^\circ \cdot \cot 47{}^\circ \cdot \cot 48{}^\circ \cdot \cot 49{}^\circ $
$ =1\cdot 1\cdot 1\cdot 1\cdot 1=1 $ $ [\because \cot 45{}^\circ =1,and,\tan \theta \cdot \cot \theta =1] $
BETA
