Complete the identify
sin(a + b)cos b - cos(a + b)sin b
Where is the equal sign?
sin (a+b) = sin a cos b + cos a sin b
cos (a+b) = cos a cos b - sin a sin b
multiply the first by cos b
multiply the second by -sin b
then add them
and you get
sin a cos^2 b + cos a sin b cos b
-cos a cos b sin b + sin a sin^2 b
low and behold that is
sin a [ cos^2 b + sin^2 b ]
but we know cos^2 + sin^2 = 1
so we end up with
sin a
That just worked out too well to be true. I wonder if I made a mistake.
Check that carefully Kelly !
To complete the identity of the expression sin(a + b)cos b - cos(a + b)sin b, we can use the trigonometric addition formulas.
The trigonometric addition formulas state:
1. sin(u + v) = sin u * cos v + cos u * sin v
2. cos(u + v) = cos u * cos v - sin u * sin v
Let's substitute u = a and v = b into the formulas.
Using formula 1, we get:
sin(a + b) = sin a * cos b + cos a * sin b
Using formula 2, we get:
cos(a + b) = cos a * cos b - sin a * sin b
Now, let's substitute these values back into the original expression sin(a + b)cos b - cos(a + b)sin b:
(sin a * cos b + cos a * sin b) * cos b - (cos a * cos b - sin a * sin b) * sin b
Expanding this expression, we get:
sin a * cos b * cos b + cos a * sin b * cos b - cos a * cos b * sin b + sin a * sin b * sin b
Next, simplify the terms:
sin a * cos^2 b + cos a * sin b * cos b - cos a * cos b * sin b + sin^2 a * sin b
Since cos^2 b is equal to (cos b)^2, we can rewrite the expression as:
sin a * (cos b)^2 + cos a * sin b * cos b - cos a * cos b * sin b + sin^2 a * sin b
This is the completed form of the given expression: sin a * (cos b)^2 + cos a * sin b * cos b - cos a * cos b * sin b + sin^2 a * sin b.