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3.Trigonometrical Ratios, Functions and Identities
medium
If $\cos A = \cos B\,\,\cos C$and $A + B + C = \pi ,$ then the value of $\cot \,B\,\cot \,C$ is
A
$1$
B
$2$
C
$\frac{1}{3}$
D
$\frac{1}{2}$
Solution
(d) We have $\cos A = \cos B\cos C$ By triangle property,
$A + B + C = \pi \Rightarrow B + C = \pi – A$
$\therefore \cos (B + C) = \cos (\pi – A) \Rightarrow \cos (B + C) = – \cos A$
$ \Rightarrow \cos B\cos C – \sin B\sin C = – \cos B\cos C$
$( \because {\rm{Given}}\cos A = \cos B\cos C)$
$ \Rightarrow 2\cos B\cos C = \sin B\sin C$
$ \Rightarrow \frac{{\cos B\cos C}}{{\sin B\sin C}} = \frac{1}{2}$
$\Rightarrow \cot B\cot C = \frac{1}{2}$.
Standard 11
Mathematics
