Validating Statements

Step 1: Use trigonometric identities.
We start with the product-to-sum identity for sine:

%5Csin%20A%20%5Csin%20B%20%3D%20%5Cfrac%7B1%7D%7B2%7D%20%5Cleft%5B%20%5Ccos(A%20-%20B)%20-%20%5Ccos(A%20%2B%20B)%20%5Cright%5D

For the given expression,

A%20%3D%20x

and

B%20%3D%20x%20%2B%20%5Cfrac%7B%5Cpi%7D%7B3%7D

, so applying the identity:

%5Csin%20x%20%5Csin%20%5Cleft(%20x%20%2B%20%5Cfrac%7B%5Cpi%7D%7B3%7D%20%5Cright)%20%3D%20%5Cfrac%7B1%7D%7B2%7D%20%5Cleft%5B%20%5Ccos%20%5Cleft(%20x%20-%20%5Cleft(%20x%20%2B%20%5Cfrac%7B%5Cpi%7D%7B3%7D%20%5Cright)%20%5Cright)%20-%20%5Ccos%20%5Cleft(%20x%20%2B%20%5Cleft(%20x%20%2B%20%5Cfrac%7B%5Cpi%7D%7B3%7D%20%5Cright)%20%5Cright)%20%5Cright%5D

This simplifies to:

%5Csin%20x%20%5Csin%20%5Cleft(%20x%20%2B%20%5Cfrac%7B%5Cpi%7D%7B3%7D%20%5Cright)%20%3D%20%5Cfrac%7B1%7D%7B2%7D%20%5Cleft%5B%20%5Ccos%20%5Cleft(%20-%20%5Cfrac%7B%5Cpi%7D%7B3%7D%20%5Cright)%20-%20%5Ccos%20%5Cleft(%202x%20%2B%20%5Cfrac%7B%5Cpi%7D%7B3%7D%20%5Cright)%20%5Cright%5D

Step 2: Find the minimum value.
We know that the cosine function has a range of [-1, 1], so the minimum value of

%5Ccos%20%5Cleft(%202x%20%2B%20%5Cfrac%7B%5Cpi%7D%7B3%7D%20%5Cright)

is -1. Therefore, the minimum value of the given expression is:

%5Cfrac%7B1%7D%7B2%7D%20%5Cleft%5B%20%5Cfrac%7B1%7D%7B2%7D%20-%20(-1)%20%5Cright%5D%20%3D%200

Final Answer: 0.

Chapter Questions
1 MCQs