Question

Proving Identities

Verify the identity$\frac{\tan x+\tan y}{\cot x+\cot y}=\tan x\tan y$

Short answer

The expression$\frac{\tan x+\tan y}{\cot x+\cot y}=\tan x\tan y$ is an identity.

Step-by-step solution

Step 1. Given information.

An expression$\frac{\tan x+\tan y}{\cot x+\cot y}=\tan x\tan y$

Step 2. Concept used.

Split given expression into two portions as LHS and RHS. Simplify both one by one and compare their outcomes. If both results are equal then given expression is an identity.

Step 3. Calculation.

Now, simplify LHS of $\frac{\tan x+\tan y}{\cot x+\cot y}=\tan x\tan y$:

$\begin{array}{l}\frac{\tan x+\tan y}{\cot x+\cot y}=\frac{\frac{\sin x}{\cos x}+\frac{\sin y}{\cos y}}{\frac{\cos x}{\sin x}+\frac{\cos y}{\sin y}}\\ =\frac{\sin x\sin y}{\cos x\cos y}\\ =\tan x\tan y\end{array}$

RHS of the equation is$\tan x\tan y$

Both RHS and LHS are equal. Hence it is an identity