express the given product as a sum containing only sines or cosines.\n\n\\( \\sin ( 9 t ) \\sin ( 7 t )…

express the given product as a sum containing only sines or cosines.\n\n\\( \\sin ( 9 t ) \\sin ( 7 t ) \\)\n\n\\( \\sin ( 9 t ) \\sin ( 7 t ) = \\)\n(simplify your answer, including any radicals. use integers or fractions for any\nnumbers in the expression.)

express the given product as a sum containing only sines or cosines.\n\n\\( \\sin ( 9 t ) \\sin ( 7 t ) \\)\n\n\\( \\sin ( 9 t ) \\sin ( 7 t ) = \\)\n(simplify your answer, including any radicals. use integers or fractions for any\nnumbers in the expression.)

Answer

Explanation:

Step1: Use the product - to - sum formula

The formula for (\sin A\sin B=\frac{1}{2}[\cos(A - B)-\cos(A + B)]). Here (A = 9U) and (B=7U). [ \begin{align*} \sin(9U)\sin(7U)&=\frac{1}{2}[\cos(9U-7U)-\cos(9U + 7U)]\ \end{align*} ]

Step2: Simplify the expressions inside the cosine functions

Simplify (9U-7U = 2U) and (9U + 7U=16U). [ \begin{align*} \sin(9U)\sin(7U)&=\frac{1}{2}[\cos(2U)-\cos(16U)]\ &=\frac{1}{2}\cos(2U)-\frac{1}{2}\cos(16U) \end{align*} ]

Answer:

(\frac{1}{2}\cos(2U)-\frac{1}{2}\cos(16U))