If A + B + C = π, Find sec A (cos B cos C − sin B sin C)

Solution

Using the identity:

\[ \cos B\cos C-\sin B\sin C = \cos(B+C) \]

Therefore,

\[ \sec A(\cos B\cos C-\sin B\sin C) = \sec A\cos(B+C) \]

Since

\[ A+B+C=\pi \]

we get

\[ B+C=\pi-A \]

Hence,

\[ \cos(B+C) = \cos(\pi-A) \]

Using the identity:

\[ \cos(\pi-\theta)=-\cos\theta \]

So,

\[ \cos(B+C)=-\cos A \]

Substituting,

\[ \sec A\cos(B+C) = \sec A(-\cos A) \]

\[ = \frac{1}{\cos A}(-\cos A) \]

\[ =-1 \]

Final Answer

\[ \boxed{ \sec A(\cos B\cos C-\sin B\sin C)=-1 } \]

Correct Option: (b)