Derive this identity from the sum and difference formulas for cosine:
sinasinb=1/2[cos(a-b)cos(a+b)]
Start with the right-hand side since it is more complex.
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To derive the identity sin(a)sin(b) = 1/2[cos(a-b)cos(a+b)], we need to start with the right-hand side of the equation and simplify it using the sum and difference formulas for cosine.
Starting with the right-hand side: 1/2[cos(a-b)cos(a+b)]
Step 1: Expand the product using the formula cos(x)cos(y) = 1/2[cos(x+y) + cos(x-y)]:
1/2[(1/2)[cos((a-b)+(a+b))] + (1/2)[cos((a-b)-(a+b))]]
Simplifying the expressions inside the brackets, we get:
1/2[(1/2)[cos(2a)] + (1/2)[cos(-2b)]]
Step 2: Simplify the cosine of a negative angle using the formula cos(-x) = cos(x):
1/2[(1/2)[cos(2a)] + (1/2)[cos(2b)]]
Step 3: Combine the fractions:
1/2[cos(2a)/2 + cos(2b)/2]
Step 4: Simplify the fractions:
1/4[cos(2a) + cos(2b)]
The right-hand side simplifies to 1/4[cos(2a) + cos(2b)].
Therefore, the identity sin(a)sin(b) = 1/2[cos(a-b)cos(a+b)] is derived from the sum and difference formulas for cosine.
To derive the identity sinasinb=1/2[cos(a-b)cos(a+b), start with the right-hand side:
1/2[cos(a-b)cos(a+b)]
Now, we can use the sum and difference formulas for cosine, which state that:
cos(a-b) = cos a cos b + sin a sin b
cos(a+b) = cos a cos b - sin a sin b
Substituting these formulas into the right-hand side, we get:
1/2[(cos a cos b + sin a sin b)(cos a cos b - sin a sin b)]
Expanding the expression, we have:
1/2[(cos a cos b)^2 - (sin a sin b)^2]
Using the identity cos^2x - sin^2x = cos(2x), we can simplify further:
1/2[cos(2a) cos(2b)]
Therefore, the derived identity is:
sin a sin b = 1/2[cos(2a) cos(2b)]
This equation is equivalent to the given identity: sin a sin b = 1/2[cos(a-b)cos(a+b)].
look at the end of your last post of this same thing
https://www.jiskha.com/display.cgi?id=1527528235