What is the value of tan 65° + 2tan 45° – 2tan 40° – tan 25°?

tan65° = cot25° = 1/tan25°, and tan40° = cot50° but simpler: tan65° – tan25° = (sin65°cos25° – sin25°cos65°)/(cos65°cos25°) = sin40°/(cos65°cos25°). Also 2tan45° = 2. And 2tan40° term. Using tan65° = cot25°: tan65° – tan25° = cot25° – tan25° = (cos²25° – sin²25°)/(sin25°cos25°) = cos50°/((1/2)sin50°) = 2cot50° = 2tan40°. So tan65° – tan25° = 2tan40°, giving tan65° – tan25° – 2tan40° = 0. Then the expression = 0 + 2tan45° = 2·1 = 2.