Question

Cos² a + cos² B - cos(a - B) cos(a + B)? ​

Answer 1

Ответ:

Применим формулы :

$\bf cosa\cdot cosb=\dfrac{1}{2}\, (\ cos(a+b)+cos(a-b)\ )\ \ ,\\\\cos^2a+sin^2a=1\ \ ,\ \ cos(-a)=cosa$                

Answer 2

To simplify the expression cos²(a) + cos²(B) - cos(a - B)cos(a + B), you can use trigonometric identities. Here's the step-by-step simplification:

1. Start with the given expression: cos²(a) + cos²(B) - cos(a - B)cos(a + B)

2. Use the trigonometric identity for the cosine of a sum of angles: cos(x + y) = cos(x)cos(y) - sin(x)sin(y)

3. Apply the identity to 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)]

4. Expand the expression: cos(a - B)cos(a + B) = cos²(a)cos²(B) - sin²(a)sin²(B)

5. Now, substitute this expanded expression back into the original expression: cos²(a) + cos²(B) - (cos²(a)cos²(B) - sin²(a)sin²(B))

6. Distribute the negative sign into the parentheses: cos²(a) + cos²(B) - cos²(a)cos²(B) + sin²(a)sin²(B)

7. Combine like terms: cos²(a) + cos²(B) - cos²(a)cos²(B) + sin²(a)sin²(B)

So, the simplified expression is: cos²(a) + cos²(B) - cos²(a)cos²(B) + sin²(a)sin²(B)