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

Note tan 65° = cot 25° = 1/tan 25° and tan 40° = cot 50°. Using tan A – cot A = –2 cot 2A: tan 25° – cot 25° = –2 cot 50° = –2 tan 40°. So tan 65° – tan 25° = cot 25° – tan 25° = 2 cot 50° = 2 tan 40°. Therefore tan 65° + 2 tan 45° – 2 tan 40° – tan 25° = (tan 65° – tan 25°) + 2 tan 45° – 2 tan 40° = 2 tan 40° + 2(1) – 2 tan 40° = 2.