Use the sum and difference identities to determine the exact value of the following expression. If the answer is undefined, write DNE. cos175°cos40°+sin175°sin40°

The given expression is in the form of the cosine of the sum of two angles. The formula for the cosine of the sum of two angles (cos(A+B)) is:

cos(A+B) = cosAcosB - sinAsinB

However, the given expression is:

cos175°cos40° + sin175°sin40°

This is similar to the formula for the cosine of the difference of two angles (cos(A-B)), which is:

cos(A-B) = cosAcosB + sinAsinB

So, we can rewrite the given expression as:

cos(175° - 40°)

This simplifies to:

cos(135°)

The exact value of cos(135°) is -√2/2. So, the exact value of the given expression is -√2/2.