e product-to-sum formula to find the exact value. Write your answer as a simplified fraction and rationalize the denominator, if necessary. \( \sin 97.5^{\circ} \cos 37.5^{\circ}=\square \)

To find the exact value of \( \sin 97.5^{\circ} \cos 37.5^{\circ} \) using the product-to-sum formula, we can use the formula: \[ \sin A \cos B = \frac{1}{2}[\sin(A+B) + \sin(A-B)] \] Given that \( A = 97.5^{\circ} \) and \( B = 37.5^{\circ} \), we can substitute these values into the formula: \[ \sin 97.5^{\circ} \cos 37.5^{\circ} = \frac{1}{2}[\sin(97.5^{\circ} + 37.5^{\circ}) + \sin(97.5^{\circ} - 37.5^{\circ})] \] Simplifying the angles inside the sine functions: \[ \sin 97.5^{\circ} \cos 37.5^{\circ} = \frac{1}{2}[\sin(135^{\circ}) + \sin(60^{\circ})] \] Now, we can calculate the exact value of \( \sin 97.5^{\circ} \cos 37.5^{\circ} \) using the values of sine for \( 135^{\circ} \) and \( 60^{\circ} \). Calculate the value by following steps: - step0: Calculate: \(\frac{1}{2}\left(\sin\left(135^{\circ}\right)+\sin\left(60^{\circ}\right)\right)\) - step1: Add the terms: \(\frac{1}{2}\times \frac{\sqrt{2}+\sqrt{3}}{2}\) - step2: Multiply the fractions: \(\frac{\sqrt{2}+\sqrt{3}}{2\times 2}\) - step3: Multiply: \(\frac{\sqrt{2}+\sqrt{3}}{4}\) The exact value of \( \sin 97.5^{\circ} \cos 37.5^{\circ} \) is \( \frac{\sqrt{2}+\sqrt{3}}{4} \).