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Chapter 5: Problem 39

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

Short Answer

Expert verified
The given equation \(\frac{\tan x+ \cot y}{\tan x \cot y}=\tan y + \cot x\) when rewritten and simplified verifies the trigonometric identity as true.

Step by step solution

01

Rewrite cotangent as 1 over tangent

Start by rewriting each \(\cot y\) as \(\frac{1}{\tan y}\) and \(\cot x\) as \(\frac{1}{\tan x}\). The formula then becomes \(\frac{\tan x + \frac{1}{\tan y}}{\tan x * \frac{1}{\tan y}} = \tan y + \frac{1}{\tan x}\)
02

Simplify the expression

Now simplify the denominator on the left side of the equation by canceling the \(\tan y\) from the numerator and the denominator, resulting in \(\frac{\tan x\cdot\tan y+1}{\tan x}=\tan y+\frac{1}{\tan x}\). We need to rewrite the numerator of the left side to display all terms in the correct order to see the similarity clearly.
03

Rearrange the equation

Rearrange the terms on the left side of the equation to get it in the format that is similar to the right side of the equation. Doing this we get \(\frac{\tan x+ \tan y}{\tan x \tan y}=\tan y + \frac{1}{\tan x}\)
04

Final Check

Observe the final format of the left side of the equation. It exactly matches the right side of the equation, confirming that the identity is correct.

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