Question

find the exact value of the expression by using appropriate identities. do not use a calculator.
\\( \sin 85 ^ { \circ } \cos 25 ^ { \circ } - \cos 85 ^ { \circ } \sin 25 ^ { \circ } \\)
\\( \sin 85 ^ { \circ } \cos 25 ^ { \circ } - \cos 85 ^ { \circ } \sin 25 ^ { \circ } = \square \\)
(type a fraction.)

Explanation:

Step1: Use the sine subtraction formula

The formula for \(\sin(A - B)=\sin A\cos B-\cos A\sin B\).
Let \(A = 85^{\circ}\) and \(B=25^{\circ}\). Then \(\sin85^{\circ}\cos25^{\circ}-\cos85^{\circ}\sin25^{\circ}=\sin(85^{\circ}- 25^{\circ})\).

Step2: Simplify the expression

\(\sin(85^{\circ}-25^{\circ})=\sin60^{\circ}\).

Step3: Recall the value of \(\sin60^{\circ}\)

We know that \(\sin60^{\circ}=\frac{\sqrt{3}}{2}\).

Answer:

\(\frac{\sqrt{3}}{2}\)