Prove the formulas for cosine of the sum and difference of two angles.

The formulas for cosine of the sum and difference of two angles can be proven using trigonometric identities.

To prove the formula for cosine of the sum of two angles, we start with the following identity:

cos(A + B) = cosA cosB - sinA sinB

We can prove this identity using the angle addition formula for cosine:

cos(A + B) = cosA cosB - sinA sinB + sinA cosB + cosA sinB
= cosA(cosB + sinB) - sinA(sinB - cosB)
= cosA cosB - sinA sinB

Now, to prove the formula for cosine of the difference of two angles, we use the identity:

cos(A - B) = cosA cosB + sinA sinB

We can prove this identity by substituting -B for B in the formula for cosine of the sum of two angles:

cos(A - B) = cosA + (-B) = cosA cos(-B) - sinA sin(-B)
= cosA cosB + sinA sinB

Therefore, we have proven the formulas for cosine of the sum and difference of two angles using trigonometric identities.