Question

which expression is equivalent to \\( \cos ( 103 ^ { \circ } ) \cos ( 54 ^ { \circ } ) - \sin ( 103 ^ { \circ } ) \sin ( 54 ^ { \circ } ) \\)?
\\( \sin ( 49 ^ { \circ } ) \\)
\\( \cos ( 49 ^ { \circ } ) \\)
\\( \sin ( 157 ^ { \circ } ) \\)
\\( \cos ( 157 ^ { \circ } ) \\)

Explanation:

Step1: Recall the cosine addition formula

The formula for \(\cos(A + B)=\cos A\cos B-\sin A\sin B\).

Step2: Identify \(A\) and \(B\)

Here \(A = 103^{\circ}\) and \(B=54^{\circ}\).
So \(\cos(103^{\circ})\cos(54^{\circ})-\sin(103^{\circ})\sin(54^{\circ})=\cos(103^{\circ}+ 54^{\circ})\).

Step3: Calculate \(A + B\)

\(103^{\circ}+54^{\circ}=157^{\circ}\).

Answer:

\(\cos(157^{\circ})\)