If $\alpha +\beta ={45}^{\circ }$ and $(\tan \alpha +1)(\tan \beta +1)=2x$, then $x$ is:

The correct answer is Option 4: 1.

Given:
$\alpha +\beta ={45}^{\circ }$
$(\tan \alpha +1)(\tan \beta +1)=2x$

Step 1: Apply tangent on both sides of the angle sum equation
Taking tangent on both sides of $\alpha +\beta ={45}^{\circ }$:

$\tan (\alpha +\beta )=\tan \left({45}^{\circ }\right)$

We know that $\tan \left({45}^{\circ }\right)=1$. Using the tangent addition formula $\tan (\alpha +\beta )=\frac{\tan \alpha +\tan \beta }{1-\tan \alpha \tan \beta }$, we get:

$\frac{\tan \alpha +\tan \beta }{1-\tan \alpha \tan \beta }=1$

Step 2: Simplify the equation
Multiplying both sides by $(1-\tan \alpha \tan \beta )$:

$\tan \alpha +\tan \beta =1-\tan \alpha \tan \beta$

Rearranging the terms:

$\tan \alpha +\tan \beta +\tan \alpha \tan \beta =1$

Step 3: Expand the given expression
Now expand $(\tan \alpha +1)(\tan \beta +1)$:

$(\tan \alpha +1)(\tan \beta +1)=\tan \alpha \tan \beta +\tan \alpha +\tan \beta +1$

Substitute $\tan \alpha +\tan \beta +\tan \alpha \tan \beta =1$ into the expanded form:

$(\tan \alpha +1)(\tan \beta +1)=1+1=2$

Step 4: Solve for $x$
We are given that $(\tan \alpha +1)(\tan \beta +1)=2x$:

$2x=2$

$x=1$