This is an important trigonometric identity known as the difference formula for cosine. It is useful in many situations, such as solving problems involving triangles with non-right angles. This identity can be derived from the addition formula for cosine:

cos(x+y) = cos(x)cos(y) – sin(x)sin(y).

To get the difference formula, we simply subtract y from both sides of the equation:

cos(x-y) = cos((x+y)-y) = cos(x+y)cos(-y) – sin(x+y)sin(-y)= cos(x)cos(y) – sin(x)sin(y).

This identity can also be derived from the addition formula for sine:

sin(x+y) = sin(x)cos(y) + cos(x)sin(y).

Again, we subtract y from both sides of the equation to get the difference formula for sine:

sin(x-y) = sin((x+y)-y) = sin(x+y)cos(-y) + cos(x+y)sin(-y)= cos(x)sin(y) – sin(x)cos(y).

Multiplying both sides of this equation by cos(x) and adding it to the equation for cos(x-y), we get the difference formula for cosine:

cos(x-y) = cos(x)cos(y) – sin(x)sin(y).

This identity can be used to solve problems involving triangles with non-right angles. For example, suppose we have a triangle with angle A equal to 60 degrees, angle B equal 30 degrees, and side c equal to 10. We can use the difference formula for cosine to find the value of side a:

a^2 = c^2 + b^2 – 2bc cosA

= 100 + 9 – 2(10)(9) cos60

= 121 – 180 cos60

= 121 – 90

= 31.

Thus, side a’s value is equal to the square root of 31, or approximately 5.566.

This identity can also solve problems involving angles that are not90 degrees. For example, suppose we have a triangle with angle A equal to 30 degrees, angle B equal to 60 degrees, and side c equal to 10. We can use the difference formula for cosine to find the value of side a:

a^2 = c^2 + b^2 – 2bc cosA

= 100 + 9 – 2(10)(9) cos30

= 121 – 180 cos30

= 121 – 90

= 31.

Thus, side a’s value is equal to the square root of 31, or approximately 5.566.