Question

If $\tan x+\tan y=\frac{5}{6}$ and $\cot x+\cot y=5$ then $\tan(x+y)$ is

• A. $\frac{6}{5}$
• B. $\frac{5}{6}$
• C. 5
• D. 6
• E. 1

Hint:

Trigonometry Tip: The relationship $\cot x + \cot y = \frac{\tan x + \tan y}{\tan x \tan y}$ is extremely common. Memorizing this shortcut lets you instantly find the product when given the sum!

Answer 1

Answer: (E)

Concept:
The addition formula for the tangent function is $\tan(x+y) = \frac{\tan x + \tan y}{1 - \tan x \tan y}$. To evaluate this, we already have the sum of the tangents, but we need to find their product using the given cotangent equation. Cotangent is the reciprocal of tangent ($\cot \theta = \frac{1}{\tan \theta}$).

Step 1: State the given equations.
We are provided with two equations: Equation 1: $\tan x + \tan y = \frac{5}{6}$ Equation 2: $\cot x + \cot y = 5$

Step 2: Convert the cotangent equation to tangents.
Rewrite Equation 2 by replacing the cotangents with the reciprocals of tangents: $$\frac{1}{\tan x} + \frac{1}{\tan y} = 5$$

Step 3: Combine the fractions.
Find a common denominator to add the fractions on the left side: $$\frac{\tan y + \tan x}{\tan x \tan y} = 5$$

Step 4: Solve for the product of the tangents.
Substitute the value from Equation 1 ($\tan x + \tan y = \frac{5}{6}$) into the numerator: $$\frac{5/6}{\tan x \tan y} = 5$$ Multiply both sides by $\tan x \tan y$ and divide by $5$: $$\tan x \tan y = \frac{5/6}{5} = \frac{1}{6}$$

Step 5: Apply the tangent addition formula.
Now plug the sum and the product into the identity for $\tan(x+y)$: $$\tan(x+y) = \frac{\tan x + \tan y}{1 - \tan x \tan y}$$ $$\tan(x+y) = \frac{5/6}{1 - 1/6}$$ $$\tan(x+y) = \frac{5/6}{5/6} = 1$$ Hence the correct answer is (E) 1.