Solving for $\tan(A+B)$ with $(1 + \tan A)(1 + \tan B) = 2$
We are given the equation $(1 + \tan A)(1 + \tan B) = 2$ and need to find the value of $\tan(A+B)$.
Step-by-step Solution
-
Expand the given equation:
$(1 + \tan A)(1 + \tan B) = 1 + \tan A + \tan B + \tan A \tan B$
So, $1 + \tan A + \tan B + \tan A \tan B = 2$
-
Simplify the equation:
Subtract 1 from both sides:
$\tan A + \tan B + \tan A \tan B = 2 - 1$
$\tan A + \tan B + \tan A \tan B = 1$
-
Rearrange the terms to match the tangent addition formula:
The formula for the tangent of the sum of two angles is:
$\tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$
From the simplified equation in step 2, rearrange to isolate $\tan A + \tan B$:
$\tan A + \tan B = 1 - \tan A \tan B$
-
Substitute into the tangent addition formula:
Divide both sides of $\tan A + \tan B = 1 - \tan A \tan B$ by $(1 - \tan A \tan B)$, assuming $1 - \tan A \tan B \neq 0$:
$\frac{\tan A + \tan B}{1 - \tan A \tan B} = \frac{1 - \tan A \tan B}{1 - \tan A \tan B}$
This simplifies to:
$\frac{\tan A + \tan B}{1 - \tan A \tan B} = 1$
-
Identify the result:
The left side of the equation is the formula for $\tan(A+B)$.
Therefore, $\tan(A+B) = 1$
The value of $\tan(A+B)$ is 1.
Similar Questions
- If $r\sin\theta = \frac{7}{2}$ and $r\cos\theta = \frac{7\sqrt{3}}{2}$, then what will be the value of r?
- If $\sin(3x - 20)^\circ = \cos(20 - 3y)^\circ$, then value of x - y will be:
- If $\sin \theta + \text{cosec } \theta = \sqrt{5}$, then the value of $\sin^3 \theta + \text{cosec}^3 \theta$ is:
- If $x + y = 75$ and $\sin x : \sin y = \frac{1}{\sqrt{2}} : \frac{1}{2}$, then $x : y$ is:
- The value of $\frac{\tan 45^\circ - \tan 30^\circ}{1 + \tan 45^\circ \tan 30^\circ}$ is :
- Angle $54^\circ$ is equivalent to (in radians):
- If $\cos x=-\frac{3}{5}$ and $x$ lies in the third quadrant, then the value of the $\sin x$ is:
- If $\sin(x - y) = \frac{\sqrt{3}}{2}$ and $\cos(x + y) = \frac{1}{2}$, where x and y are positive acute angles and $x \ge y$, then the value of $x$ is:
- If $\sin\theta - \cos\theta = \frac{\sqrt{3}}{2}$, then find the positive value of $\sin\theta + \cos\theta$.
- If $\sin \theta + \cos \theta = \frac{\sqrt{7}}{2}$, then find the value of $\sin \theta - \cos \theta$.
Important Questions from Trigonometry
The given equation can be reduced to
If sin2x = a - b√c, where a and b are natural numbers and c is prime number, then what is the value of a - b + 2c ?
Let θ be a positive angle. If the number of degrees in θ is divided by the number of radians in θ, then an irrational number 180 / π results. If the number of degrees in θ is multiplied by the number of radians in θ, then an irrational number 125π / 9 results. The angle θ must be equal to
What is sin 2α equal to?
If \(\sin θ = \frac{8}{{17}}\) , then find the value of tan θ.
