Can you show steps on deriving the identity from the sum and difference formulas for cosine:
sin a sin b = (1 / 2)[cos(a – b) – cos(a + b)]
RS = (1/2)(cosacosb + sinasinb - (cosacosb - sinasinb)
= (1/2)(2sinasinb)
= sinasinb
= LS
Certainly! I will guide you through the steps to derive the given identity using the sum and difference formulas for cosine.
The sum and difference formulas for cosine are:
1. cos(a + b) = cos a cos b - sin a sin b
2. cos(a - b) = cos a cos b + sin a sin b
Now, let's start with the left-hand side (LHS) of the identity you provided:
LHS = sin a sin b
To simplify this expression, we can rewrite it using the identity:
sin x = (1/2) [cos(x - y) - cos(x + y)]
where x = a + b and y = a - b.
Therefore, the LHS can be rewritten as:
LHS = (1/2) [cos(a + b - a + b) - cos(a + b + a - b)]
Simplifying further:
LHS = (1/2) [cos(2b) - cos(2a)]
Now, let's move on to the right-hand side (RHS) of the identity:
RHS = (1 / 2)[cos(a – b) – cos(a + b)]
To match the RHS to the expression we obtained for the LHS, we need to make the following substitutions:
- Substitute a with (a + b)
- Substitute b with (a - b)
RHS = (1 / 2)[cos((a + b) – (a - b)) – cos((a + b) + (a - b))]
Simplifying the RHS expression:
RHS = (1 / 2)[cos(2b) – cos(2a)]
As you can see, the LHS and RHS expressions are now identical:
LHS = (1/2) [cos(2b) - cos(2a)]
RHS = (1 / 2)[cos(2b) – cos(2a)]
Therefore, we have successfully derived the given identity using the sum and difference formulas for cosine.