Question

Sin(a+b)cosa-cos(a+b)sina

Answer 1

Упрощаем выражение, используя sin(t)cos(s)-cos(t)sin(s)=sin(t-s), получается sin(a+b-a)=sin(b)

Answer 2

To simplify the expression, we can use the trigonometric identity for the sine of a sum:

sin(a + b) = sin(a) * cos(b) + cos(a) * sin(b)

Now, let's plug this identity into the given expression:

sin(a + b) * cos(a) - cos(a + b) * sin(a)

Replace sin(a + b) using the identity:

(sin(a) * cos(b) + cos(a) * sin(b)) * cos(a) - cos(a + b) * sin(a)

Next, use the trigonometric identity for the cosine of a sum:

cos(a + b) = cos(a) * cos(b) - sin(a) * sin(b)

Replace cos(a + b) in the expression:

(sin(a) * cos(b) + cos(a) * sin(b)) * cos(a) - (cos(a) * cos(b) - sin(a) * sin(b)) * sin(a)

Now, distribute the terms:

sin(a) * cos(b) * cos(a) + cos(a) * sin(b) * cos(a) - cos(a) * cos(b) * sin(a) + sin(a) * sin(b) * sin(a)

Combine like terms:

cos(b) * cos(a) * sin(a) - cos(b) * sin(a) * cos(a) + sin(a) * sin(b) * sin(a)

Now, factor out sin(a):

sin(a) * (cos(b) * cos(a) - cos(b) * sin(a) + sin(b) * sin(a))

And that's the simplified expression:

sin(a + b) * cos(a) - cos(a + b) * sin(a) = sin(a) * (cos(b) * cos(a) - cos(b) * sin(a) + sin(b) * sin(a))