Question

If A = ${20}^{\circ }$ and B = ${25}^{\circ }$, Find the value of tanA + tanB + tanAtanB.

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Answer 1

We have to find tanA + tanB + tanAtanB.
tanAtanB is one term and tanA and tanB are the other terms. By observing this much, we may

be able to guess the formula / expansion we are going to use.
We are also given A = ${20}^{\circ }$ and B = ${25}^{\circ }$. More than individual values, the sum of them is important here. A+ B = ${45}^{\circ }$ Using these two observations, we can easily guess that we are going to use.
tan (A+B).
$\Rightarrow$ tan (A + B) = $\frac{tanA+tanB}{1-tanAtanB}$
A+B= ${45}^{\circ }$
$\Rightarrow$ tan ${45}^{\circ }$ = 1 = $\frac{tanA+tanB}{1-tanAtanB}$

$\Rightarrow$ 1 - tanAtanB = tanA + tanB
$\Rightarrow$ tanA + tanB + tanAtanB = 1

If we think of taking a common factor from two terms in the expression, you will see that it

does not simplify the terms / does not give any insight to further steps.
Key steps/concepts: (1) Expansion of tan (A+B)
(2)Guessing the method after seeing the terms tanA, tanB and tan (A+B).