find the formula for tan(x-y) in terms of tanx) and tan(y)

To find the formula for \(\tan(x - y)\) in terms of \(\tan(x)\) and \(\tan(y)\), we can use the angle subtraction identity for the tangent function.

The angle subtraction formula for tangent is:

\[
\tan(x - y) = \frac{\tan(x) - \tan(y)}{1 + \tan(x) \tan(y)}
\]

Here's the derivation step-by-step:

1. Recall the sine and cosine subtraction identities:
   \[
   \sin(x - y) = \sin(x)\cos(y) - \cos(x)\sin(y)
   \]
   \[
   \cos(x - y) = \cos(x)\cos(y) + \sin(x)\sin(y)
   \]

2. By definition, \(\tan(x) = \frac{\sin(x)}{\cos(x)}\) and \(\tan(y) = \frac{\sin(y)}{\cos(y)}\).

3. Substitute these into the tangent formula \(\tan(x - y) = \frac{\sin(x - y)}{\cos(x - y)}\):
   \[
   \tan(x - y) = \frac{\sin(x)\cos(y) - \cos(x)\sin(y)}{\cos(x)\cos(y) + \sin(x)\sin(y)}
   \]

4. Divide the numerator and the denominator by \(\cos(x)\cos(y)\):
   \[
   \tan(x - y) = \frac{\frac{\sin(x)\cos(y)}{\cos(x)\cos(y)} - \frac{\cos(x)\sin(y)}{\cos(x)\cos(y)}}{\frac{\cos(x)\cos(y)}{\cos(x)\cos(y)} + \frac{\sin(x)\sin(y)}{\cos(x)\cos(y)}}
   \]

5. Simplify each term:
   \[
   \tan(x - y) = \frac{\frac{\sin(x)}{\cos(x)} - \frac{\sin(y)}{\cos(y)}}{1 + \frac{\sin(x)\sin(y)}{\cos(x)\cos(y)}}
   \]
   \[
   \tan(x - y) = \frac{\tan(x) - \tan(y)}{1 + \tan(x) \tan(y)}
   \]

Therefore, the formula for \(\tan(x - y)\) in terms of \(\tan(x)\) and \(\tan(y)\) is:

\[
\tan(x - y) = \frac{\tan(x) - \tan(y)}{1 + \tan(x) \tan(y)}
\]

Would you like more details or have any questions?

Here are 8 related questions you might find interesting:
1. What is the formula for \(\tan(x + y)\)?
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Explore the formula for \( \tan(x - y) \) in terms of \( \tan(x) \) and \( \tan(y) \). Learn how to derive and apply the angle subtraction identity for the tangent function with detailed step-by-step instructions.

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