If x+y=π/4, then prove (1+tanx)(1+tany)=2

In this post, we will prove (1+tanx)(1+tany)=2 when x+y is equal to π/4 =45°. To prove this, we will use the following formula:

tan(x+y) = $\dfrac{\tan x+\tan y}{1-\tan x \tan y}$.

Question: If x+y=π/4, then prove that (1+tanx)(1+tany)=2.

Solution:

Step 1:

Given that x+y=π/4.

Therefore,

tan(x+y) = tan(π/4)

⇒ $\dfrac{\tan x+\tan y}{1-\tan x \tan y}$ = 1

Step 2:

Cross-multiplying, we get tanx + tany = 1- tanx tany

⇒ tanx + tany + tanx tany = 1

Step 3:

Adding 1 to both sides, we obtain that

1+tanx + tany + tanx tany = 1+1

⇒ (1+tanx) + tany (1+ tanx) = 2

⇒ (1+tanx) (1+ tany) = 2.

Thus, we have shown that if x+y=π/4, then (1+tanx) (1+ tany) = 2.

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