Question

Cos a cosß-cos(a+b)cos(a - b) - sin a sinß​

Answer 1

Пошаговое объяснение:

$cosa\cdot cos\beta-cos(a-\beta )=cosa\cdot cos\beta -(cosa\cdot cos\beta -sina\cdot sin\beta )=\\\\=sina\cdot sin\beta \\\\\\cos(a-\beta )-sina\cdot sin\beta =(cosa\cdot cos\beta +sina\cdot sin\beta )-sina\cdot sin\beta =\\\\=cosa\cdot cos\beta$

Answer 2

Ответ:

Пошаговое объяснение:

cos a * cos b - cos (a+b) = 1/2 ( cos (a+b) + cos (a-b) ) - cos (a+b) = 1/2 ( cos (a-b) - cos (a+b) ) = sin a * sin b

cos (a-b) - sin a * sin b = cos (a-b) - 1/2 ( cos (a-b) - cos (a+b) ) = 1/2 ( cos (a-b) + cos (a+b) ) = cos a * cos b

Answer 3

To simplify the expression "cos(a)cos(b) - cos(a+b)cos(a-b) - sin(a)sin(b)", we'll use some trigonometric identities. One of the most useful identities for this expression is the cosine double-angle identity:

cos(2θ) = cos^2(θ) - sin^2(θ)

Now, let's apply the double-angle identity to simplify the expression step by step:

1. cos(a)cos(b) - cos(a+b)cos(a-b) - sin(a)sin(b)

2. Now, we can use the sum-to-product identities to simplify cos(a+b)cos(a-b):

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

3. Now, let's substitute this back into the original expression:

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

4. Expand the expression inside the brackets:

cos(a)cos(b) - (cos^2(a)cos^2(b) - sin^2(a)sin^2(b)) - sin(a)sin(b)

5. Now, use the identity cos^2(θ) - sin^2(θ) = cos(2θ):

cos(a)cos(b) - (cos^2(a)cos^2(b) - (1-cos^2(a))(1-cos^2(b))) - sin(a)sin(b)

6. Expand the squared terms:

cos(a)cos(b) - (cos^2(a)cos^2(b) - (1 - 2cos^2(a) + cos^4(a))(1 - 2cos^2(b) + cos^4(b))) - sin(a)sin(b)

7. Distribute the negative sign inside the brackets:

cos(a)cos(b) - cos^2(a)cos^2(b) + (1 - 2cos^2(a) + cos^4(a))(1 - 2cos^2(b) + cos^4(b)) - sin(a)sin(b)

8. Expand the terms inside the second brackets:

cos(a)cos(b) - cos^2(a)cos^2(b) + (1 - 2cos^2(a) + cos^4(a) - 2cos^2(b) + 4cos^2(a)cos^2(b) - 2cos^4(a) + cos^4(a)cos^4(b)) - sin(a)sin(b)

9. Combine like terms:

cos(a)cos(b) - cos^2(a)cos^2(b) + 1 - 2cos^2(a) - 2cos^2(b) + 3cos^4(a)cos^2(b) - 2cos^4(a) + cos^4(a)cos^4(b) - sin(a)sin(b)

And there you have the simplified expression:

1 - 2cos^2(a) - 2cos^2(b) + 3cos^4(a)cos^2(b) - 2cos^4(a) + cos^4(a)cos^4(b) - cos^2(a)cos^2(b) + cos(a)cos(b) - sin(a)sin(b)

You can further simplify this expression if you have specific values for 'a' and 'b' or if you want to apply additional trigonometric identities.