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Which expression is equivalent to $$\frac{\tan{2x} - \tan{x}}{1 + \tan{2x} \tan{x}}$$? - HSC - SSCE Mathematics Extension 1 - Question 3 - 2016 - Paper 1

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Which-expression-is-equivalent-to---$$\frac{\tan{2x}---\tan{x}}{1-+-\tan{2x}-\tan{x}}$$?-HSC-SSCE Mathematics Extension 1-Question 3-2016-Paper 1.png

Which expression is equivalent to $$\frac{\tan{2x} - \tan{x}}{1 + \tan{2x} \tan{x}}$$?

Worked Solution & Example Answer:Which expression is equivalent to $$\frac{\tan{2x} - \tan{x}}{1 + \tan{2x} \tan{x}}$$? - HSC - SSCE Mathematics Extension 1 - Question 3 - 2016 - Paper 1

Step 1

Identify the given expression

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Answer

We start with the expression tan⁡2x−tan⁡x1+tan⁡2xtan⁡x\frac{\tan{2x} - \tan{x}}{1 + \tan{2x} \tan{x}}.

Step 2

Apply the tangent subtraction formula

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Answer

We can use the formula for the tangent of a difference: tan⁡a−b=tan⁡a−tan⁡b1+tan⁡atan⁡b\tan{a - b} = \frac{\tan{a} - \tan{b}}{1 + \tan{a} \tan{b}}. Here, let ( a = 2x ) and ( b = x ). Thus, we have:

tan⁡(2x−x)=tan⁡x\tan{(2x - x)} = \tan{x}.

This shows that the expression simplifies directly to ( \tan{x} ).

Step 3

Select the correct answer

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Answer

The correct expression equivalent to the given expression is (A) (\tan{x}).

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