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

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

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

tan(A + B) = sin(A + B) / cos(A + B)

Using the sum identities for sine and cosine, we can rewrite this as:

tan(A + B) = (sinA cosB + cosA sinB) / (cosA cosB - sinA sinB)

Next, we simplify the numerator and denominator by factoring out sinA and cosA respectively:

tan(A + B) = (sinA (cosB + sinB tanA)) / (cosA (cosB - sinB tanA))

Finally, we use the identity tanA = sinA / cosA to simplify the expression:

tan(A + B) = (tanA + tanB) / (1 - tanA tanB)

This is the formula for tangent of the sum of two angles.

To prove the formula for tangent of the difference of two angles, we start with the identity:

tan(A - B) = sin(A - B) / cos(A - B)

Using the difference identities for sine and cosine, we can rewrite this as:

tan(A - B) = (sinA cosB - cosA sinB) / (cosA cosB + sinA sinB)

Next, we simplify the numerator and denominator by factoring out sinA and cosA respectively:

tan(A - B) = (sinA (cosB - sinB tanA)) / (cosA (cosB + sinB tanA))

Finally, we use the identity tanA = sinA / cosA to simplify the expression:

tan(A - B) = (tanA - tanB) / (1 + tanA tanB)

This is the formula for tangent of the difference of two angles.