Question

If \(x+y = \frac{π}{4}\), then \((1+tanx)(1+tany) =\)

• A. \(1\)
• B. \(2\)
• C. \(3\)
• D. \(0\)

Hint:

Expand tan(x+y) = 1 and substitute into the product.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
We use the sum formula for tangent and the fact that \(\tan\frac{\pi}{4}=1\).

Step 2: Key Formula or Approach
\[ \tan(x+y)=\frac{\tan x+\tan y}{1-\tan x\tan y} \]

Step 3: Use the given condition
\[ \frac{\tan x+\tan y}{1-\tan x\tan y}=\tan\frac{\pi}{4}=1 \]
\[ \tan x+\tan y=1-\tan x\tan y \]

Step 4: Expand the product
\[ (1+\tan x)(1+\tan y)=1+\tan x+\tan y+\tan x\tan y \]
Replace \(\tan x+\tan y\) by \(1-\tan x\tan y\):
\[ =1+(1-\tan x\tan y)+\tan x\tan y=2 \]
So the value is 2, option (B).

Final Answer:
The product is always 2 when x + y = pi/4, option (B). \[ \boxed{2} \]