Question

simplify the expression by using an appropriate identity. do not use a calculator
$(cos 59^{circ })(cos 31^{circ })-(sin 59^{circ })(sin 31^{circ })$
write the expression using the sum or difference formula for sine or cosine. choose the correct answer below
a. $(cos 59^{circ })(cos 31^{circ })-(sin 59^{circ })(sin 31^{circ })=cos (59^{circ }-31^{circ })=cos (28^{circ })$
b. $(cos 59^{circ })(cos 31^{circ })-(sin 59^{circ })(sin 31^{circ })=cos (59^{circ }+31^{circ })=cos (90^{circ })$
c. $(cos 59^{circ })(cos 31^{circ })-(sin 59^{circ })(sin 31^{circ })=sin (-59^{circ }+(-31^{circ }))=sin (-90^{circ })$
d. $(cos 59^{circ })(cos 31^{circ })-(sin 59^{circ })(sin 31^{circ })=sin (59^{circ }+31^{circ })=sin (90^{circ })$
the exact value of $(cos 59^{circ })(cos 31^{circ })-(sin 59^{circ })(sin 31^{circ })$ is
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. rationalize all denominators.

Explanation:

Step1: Recall the cosine sum formula

The formula for \(\cos(A + B)=\cos A\cos B-\sin A\sin B\). In the given expression \((\cos59^{\circ})(\cos31^{\circ})-(\sin59^{\circ})(\sin31^{\circ})\), we can let \(A = 59^{\circ}\) and \(B=31^{\circ}\).

Step2: Apply the formula

By the formula \(\cos(A + B)=\cos A\cos B-\sin A\sin B\), when \(A = 59^{\circ}\) and \(B = 31^{\circ}\), we have \((\cos59^{\circ})(\cos31^{\circ})-(\sin59^{\circ})(\sin31^{\circ})=\cos(59^{\circ}+ 31^{\circ})\).

Step3: Calculate the value

\(\cos(59^{\circ}+31^{\circ})=\cos(90^{\circ})\). Since \(\cos(90^{\circ}) = 0\)

Answer:

B. \((\cos59^{\circ})(\cos31^{\circ})-(\sin59^{\circ})(\sin31^{\circ})=\cos(59^{\circ}+31^{\circ})=\cos(90^{\circ})\)

The exact value of \((\cos59^{\circ})(\cos31^{\circ})-(\sin59^{\circ})(\sin31^{\circ})\) is \(0\)