To prove that sin(πx) sin(x) is equal to a certain expression, we can use the product-to-sum identities in trigonometry.
The product-to-sum identities state that:
- sin(a) sin(b) = (1/2) [cos(a-b) – cos(a+b)]
In our case, let a = πx and b = x. Then we can rewrite the expression:
sin(πx) sin(x) = (1/2) [cos(πx – x) – cos(πx + x)]
Now, simplify the arguments of the cosine functions:
- πx – x = (π – 1)x
- πx + x = (π + 1)x
This gives us:
sin(πx) sin(x) = (1/2) [cos((π – 1)x) – cos((π + 1)x)]
Thus, we have proved that:
sin(πx) sin(x) = (1/2) [cos((π – 1)x) – cos((π + 1)x)]
This shows the relationship between the product of the sine functions and the sum of cosines.