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