To determine which expression is equivalent to [tex]\(\cos(103^\circ) \cos(54^\circ) - \sin(103^\circ) \sin(54^\circ)\)[/tex], we can use the angle addition formula for cosine. The angle addition formula states that:
[tex]\[
\cos(A + B) = \cos(A)\cos(B) - \sin(A)\sin(B)
\][/tex]
Given the expression [tex]\(\cos(103^\circ) \cos(54^\circ) - \sin(103^\circ) \sin(54^\circ)\)[/tex], we can recognize this as matching the form of [tex]\(\cos(A + B)\)[/tex] with [tex]\(A = 103^\circ\)[/tex] and [tex]\(B = 54^\circ\)[/tex]:
[tex]\[
\cos(103^\circ) \cos(54^\circ) - \sin(103^\circ) \sin(54^\circ) = \cos(103^\circ + 54^\circ)
\][/tex]
Next, we add the angles:
[tex]\[
103^\circ + 54^\circ = 157^\circ
\][/tex]
Thus, the given expression simplifies to:
[tex]\[
\cos(157^\circ)
\][/tex]
Therefore, the expression that is equivalent to [tex]\(\cos(103^\circ) \cos(54^\circ) - \sin(103^\circ) \sin(54^\circ)\)[/tex] is:
[tex]\[
\cos(157^\circ)
\][/tex]
So the correct answer is:
[tex]\[
\cos(157^\circ)
\][/tex]
