write the expression as the sine, cosine, or tangent of an angle. then find the exact value of the…

write the expression as the sine, cosine, or tangent of an angle. then find the exact value of the expression.\n\\( \\sin \\frac { 7 \\pi } { 12 } \\cos \\frac { \\pi } { 12 } - \\cos \\frac { 7 \\pi } { 12 } \\sin \\frac { \\pi } { 12 } \\)\nwrite the expression as the sine, cosine, or tangent of a single angle.\n\\( \\sin \\frac { 7 \\pi } { 12 } \\cos \\frac { \\pi } { 12 } - \\cos \\frac { 7 \\pi } { 12 } \\sin \\frac { \\pi } { 12 } = \\square \\)\n(use integers or fractions for any numbers in the expression. simplify your answer. do not evaluate.)

write the expression as the sine, cosine, or tangent of an angle. then find the exact value of the expression.\n\\( \\sin \\frac { 7 \\pi } { 12 } \\cos \\frac { \\pi } { 12 } - \\cos \\frac { 7 \\pi } { 12 } \\sin \\frac { \\pi } { 12 } \\)\nwrite the expression as the sine, cosine, or tangent of a single angle.\n\\( \\sin \\frac { 7 \\pi } { 12 } \\cos \\frac { \\pi } { 12 } - \\cos \\frac { 7 \\pi } { 12 } \\sin \\frac { \\pi } { 12 } = \\square \\)\n(use integers or fractions for any numbers in the expression. simplify your answer. do not evaluate.)

Answer

Explanation:

Step1: Use the sine subtraction formula

The formula for (\sin(A - B)=\sin A\cos B-\cos A\sin B). Here (A = \frac{7\pi}{12}) and (B=\frac{\pi}{12}). So, (\sin\frac{7\pi}{12}\cos\frac{\pi}{12}-\cos\frac{7\pi}{12}\sin\frac{\pi}{12}=\sin\left(\frac{7\pi}{12}-\frac{\pi}{12}\right))

Step2: Simplify the angle

(\frac{7\pi}{12}-\frac{\pi}{12}=\frac{7\pi - \pi}{12}=\frac{6\pi}{12}=\frac{\pi}{2})

Answer:

(\sin\frac{\pi}{2})