Question

Cos 5x cos 4x + sin 5x * sin 4x = 1

Answer 1

Ответ:

Решение в файле

Answer 2

The expression you provided is a trigonometric identity that can be proven using various trigonometric identities. In this case, you can use the product-to-sum identities to simplify the expression. The product-to-sum identities are:

1. cos(A) * cos(B) = (1/2) * [cos(A + B) + cos(A - B)]
2. sin(A) * sin(B) = (1/2) * [cos(A - B) - cos(A + B)]

Let's apply these identities to your expression:

Cos(5x) * Cos(4x) + Sin(5x) * Sin(4x)

= (1/2) * [Cos(5x + 4x) + Cos(5x - 4x)] - (1/2) * [Cos(5x - 4x) - Cos(5x + 4x)]

Now, you can see that the terms with opposite signs in the second pair cancel out:

= (1/2) * [Cos(9x) + Cos(x)] - (1/2) * [Cos(x) - Cos(9x)]

Now, you can combine like terms:

= (1/2) * Cos(9x) + (1/2) * Cos(x) - (1/2) * Cos(x) + (1/2) * Cos(9x)

The two middle terms with opposite signs cancel out:

= (1/2) * Cos(9x) + (1/2) * Cos(9x)

Now, you have:

= (1/2) * Cos(9x) + (1/2) * Cos(9x) = Cos(9x)

So, the expression simplifies to:

Cos(9x)

Therefore, Cos(5x) * Cos(4x) + Sin(5x) * Sin(4x) is equal to 1, and the simplified expression is Cos(9x).