Help with trigonometry equation

Question

My niece came to me with a math question for her schoolbooks: Proof the equation below, when you know a + b = 45°: (1+tan(a))(1+tan(b)) = 2

It seemed a pretty easy equation to me so I thought I'd give it a go. Ended up changing the 1 with tan(a+b) and then using tan(a+b)=(tan(a)+tan(b))/(1-tan(a).tan(b)) , but that kinda got me stuck...

Anyone knows what the correct approach is here?

Answer 1

It's not an equation, since there's no unknown. It's only a formula to prove.

Hint:

Expand $(1+\tan a)(1+\tan b)$ and use the addition formula for the tangent: $$\tan(a+b)=\frac{\tan a+\tan b}{1-\tan a\,\tan b}.$$

Answer 2

You have\begin{align}(1+\tan a)(1+\tan b)&=(1+\tan a)\bigl(1+\tan(45^\circ-a)\bigr)\\&=(1+\tan a)\left(1+\frac{1-\tan a}{1+\tan a}\right)\\&=1+\tan(a)+1-\tan(a)\\&=2.\end{align}