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Question
If A + B + C = π, then sec A (cos B cos C − sin B sin C) is equal to
Options
(a) 0
(b) −1
(c) 1
(d) None of these
Solution
(b) −1
π = 180°
Now, using the identities
\[\cos\left( A + B \right) = \cos A\cos B - \sin A\sin B\] and \[\sin\left( A + B \right) = \sin A\cos B + \cos A\sin B\]
\[\sec A\left( \cos B\cos C - \sin B\sin C \right) = \frac{- \cos A\cos B^2 + \cos B\sin A\sin B - \sin B\sin A\cos B - \sin^2 B\cos A}{\cos A}\]
\[\Rightarrow \sec A\left( \cos B\cos C - \sin B\sin C \right) = \frac{- \cos A\left( \cos^2 B + \sin^2 B \right)}{\cos A}\]
\[ \Rightarrow \sec A\left( \cos B\cos C - \sin B\sin C \right) = \frac{- \cos A}{\cos A} = - 1\]
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