If x1, x2 ∈ℝ are such that none of x1, x2, and x1+x2 equals (2 k+1) π/ 2 for any k ∈ℤ, then show

step 1 We have to prove that tan inverse of x plus tan inverse of 1 divided by x is equal to pi divided

If x1, x2 ∈ℝ are such that none of x1, x2, and x1+x2 equals...

If $x_{1}, x_{2} \in \mathbb{R}$ are such that none of $x_{1}, x_{2}$, and $x_{1}+x_{2}$ equals $(2 k+1) \pi / 2$ for any $k \in \mathbb{Z}$, then show that
$$
\tan \left(x_{1}+x_{2}\right)=\frac{\tan x_{1}+\tan x_{2}}{1-\tan \left(x_{1}+x_{2}\right)}.
$$

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Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variables (within their domains). They are fundamental tools in trigonometry that simplify the manipulation and solving of trigonometric expressions and equations, offering a way to express complex relations in a more manageable form.

Tangent Addition Formula

The tangent addition formula, given by tan(a + b) = (tan a + tan b) / (1 - tan a tan b), is an essential identity in trigonometry. Derived from the sine and cosine addition formulas, it allows the tangent of a sum of two angles to be expressed directly in terms of the tangents of the individual angles, thereby facilitating the solution of many problems involving tangent functions.

Domain Restrictions in Trigonometric Functions

In trigonometric identities, especially those involving the tangent function, it is crucial to consider domain restrictions because functions like tangent are undefined when their corresponding cosine values are zero. Ensuring that values such as angles or sums of angles do not result in division by zero (for instance by avoiding angles of the form (2k+1)?/2) is essential for the correct application of these identities.

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