Verifying the Identity
To verify the identity cos(x) * cos(y) + cos(x) * cos(y) – 2 * sin(x) * sin(y), we first need to simplify the expression.
1. Start with the left-hand side:
cos(x) * cos(y) + cos(x) * cos(y)2. Combine like terms:
2 * cos(x) * cos(y)3. Now, we have:
2 * cos(x) * cos(y) - 2 * sin(x) * sin(y)4. Factor out the common factor of 2:
2 (cos(x) * cos(y) - sin(x) * sin(y))5. Recognizing the cosine of a sum identity:
According to the cosine addition formula:
cos(a + b) = cos(a) * cos(b) - sin(a) * sin(b)We can see that our expression fits this format with a = x and b = y.
6. Therefore:
cos(x + y) = cos(x) * cos(y) - sin(x) * sin(y)7. Putting it all together, we conclude that:
2 * (cos(x + y))Thus, we can verify that:
2 * (cos(x) * cos(y) - sin(x) * sin(y)) = cos(x) * cos(y) + cos(x) * cos(y) - 2 * sin(x) * sin(y)Conclusion
After simplifying and applying the cosine identity, we see that the original expression is indeed valid. Therefore, we can confirm that the identity holds true.