find the exact value of the expression by using appropriate identities. do not use a calculator.\n\\( \\sin…

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

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

Answer

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})