Question

If A + C = B, then tan A tan B tan C =
A) tan A + tan B + tan C
B) tan B – tan C – tan A
C) tan A + tan B – tan C
D) – (tan A + tan B + tan C)

Answer 1

Hint: In this problem we will use the trigonometric formula of tan (A+B). Then we will use the given condition (A + C = B) to get the solution.

Complete step-by-step answer:
Now, it is given that A + C = B, so we will use this condition to find the solution of the given problem.
Now, $\tan {\text{ (A + B) = }}\dfrac{{{\text{tan A + }}{\text{ tan B}}}}{{1{\text{ }} - {\text{ tan A tan B}}}}$
Now in the above property we will put the values, A = A and B = C.so, we get
$\tan {\text{ (A + C) = }}\dfrac{{{\text{tan A + }}{\text{ tan C}}}}{{1{\text{ }} - {\text{ tan A tan C}}}}$ As, A + C = B, so the above equation can be written as
$\tan {\text{ B = }}\dfrac{{{\text{tan A + }}{\text{ tan C}}}}{{1{\text{ }} - {\text{ tan A tan C}}}}$
Now cross- multiplying both sides, we get
$\tan {\text{ B(1 - tan A tan C) = tan A + tan C}}$
$\tan {\text{ B - tan A tan B tan C = tan A + tan C}}$
${\text{tan A tan B tan C = tan B - tan A - tan C}}$
Which can be written as ${\text{tan A tan B tan C = tan B - tan C - tan A}}$.
So, option (B) is the correct answer.

Note: Such problems look difficult but they are very easy to solve. You just have to apply a property and use the condition given in the question. By applying the condition in the property, you will get the correct answer by doing a simple calculation. Just make sure that you use proper identity and do all the calculations correctly.