[tex]\cos (5 \pi) \sin (8 \pi)=\frac{1}{2}[\sin 13 \pi+\sin (3 \pi)][/tex] is the answer.
Explanation:
To write the product [tex]\cos (5 \pi) \sin (8 \pi)[/tex] as the sum using product-sum identities.
The product-sum identity for [tex]cos A sinB[/tex] is given by
[tex]\cos A \sin B=\frac{1}{2}[\sin (A+B)-\sin (A-B)][/tex]
Now, we shall substitute the value for A and B in this formula.
Thus, [tex]A=5 \pi[/tex] and [tex]B=8 \pi[/tex], we have,
[tex]\cos (5 \pi) \sin (8 \pi)=\frac{1}{2}[\sin (5 \pi+8 \pi)+\sin (5 \pi-8 \pi)][/tex]
Adding the terms within the bracket,
[tex]\cos (5 \pi) \sin (8 \pi)=\frac{1}{2}[\sin 13 \pi-\sin (-3 \pi)][/tex]
Since, we know that [tex]\sin (-x)=-\sin (x)[/tex], we have,
[tex]\cos (5 \pi) \sin (8 \pi)=\frac{1}{2}[\sin 13 \pi+\sin (3 \pi)][/tex]
Thus, the solution is [tex]\cos (5 \pi) \sin (8 \pi)=\frac{1}{2}[\sin 13 \pi+\sin (3 \pi)][/tex]
