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3.Trigonometrical Ratios, Functions and Identities
medium
જો $A + B + C = {180^o},$ તો $(\cot B + \cot C)$ $(\cot C + \cot A)\,\,(\cot A + \cot B) = . . . .$
A
$\sec A\,\sec B\,\sec C$
B
${\rm{cosec}}\,A\,{\rm{cosec}}\,B\,{\rm{cosec}}\,C$
C
$\tan A\,\tan B\,\tan C$
D
$1$
Solution
(b) $\cot B + \cot C = \frac{{\sin C\,\cos B + \sin B\,\cos C}}{{\sin B\,\sin C}}$
$ = \frac{{\sin (B + C)}}{{\sin B\,\sin C}}$
$ = \frac{{\sin ({{180}^o} – A)}}{{\sin B\,\sin C}}$
$ = \frac{{\sin A}}{{\sin B\sin C}}$
Similarly, $\cot C + \cot A = \frac{{\sin B}}{{\sin C\sin A}}$
and $\cot A + \cot B = \frac{{\sin C}}{{\sin A\sin B}}$
Therefore, $(\cot B + \cot C)(\cot C + \cot A)(\cot A + \cot B)$
$ = \frac{{\sin A}}{{\sin B\sin C}}.\frac{{\sin B}}{{\sin C\sin A}}.\frac{{\sin C}}{{\sin A\sin B}}$
$ = \cos {\rm{ec}}A\cos {\rm{ec}}B\cos {\rm{ec}}C.$
Standard 11
Mathematics
