Find the exact value of the expression.

(Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.

Ask by Gough Spencer. in the United States

Jan 23,2025

Question

Find the exact value of the expression.

(Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.

Ask by Gough Spencer. in the United States

Jan 23,2025

UpStudy AI · Accepted Answer

To find the exact value of the expression:

$$
\cos 5^{\circ} \cos 25^{\circ} - \sin 5^{\circ} \sin 25^{\circ}
$$

We can use the **cosine addition formula**, which states:

$$
\cos(A + B) = \cos A \cos B - \sin A \sin B
$$

Let $$ A = 5^{\circ} $$ and $$ B = 25^{\circ} $$. Applying the formula:

$$
\cos 5^{\circ} \cos 25^{\circ} - \sin 5^{\circ} \sin 25^{\circ} = \cos(5^{\circ} + 25^{\circ}) = \cos 30^{\circ}
$$

We know the exact value of $$ \cos 30^{\circ} $$ from trigonometric identities:

$$
\cos 30^{\circ} = \frac{\sqrt{3}}{2}
$$

**Final Answer:**

$$
\frac{\sqrt{3}}{2}
$$
$$ \frac{\sqrt{3}}{2} $$