Question

Analyze this derivation of the tangent double angle identity. tan(2x) = tan (x + x) Equals StartFraction tangent (x) + tangent (x) Over 1 minus tangent (x) tangent (x) EndFraction Equals StartFraction 2 tangent (x) Over 1 minus tangent squared (x) EndFraction Choose the justification for each step. Step 1: . Step 2: . Step 3: .

Answer 1

The derivation steps and the reasons are:

• $\tan(2x) = \tan(x+x)$ -------- Addition
• $\tan(2x) = \frac{\tan(x) + \tan(x)}{1 - \tan(x) \tan(x)}$ --------- Tangent sum identity
• $\tan(2x) = \frac{\tan(x) + \tan(x)}{1 - \tan^2(x)}$ ------- Simplify

What is the tangent sum identity?

The tangent sum identity states that:

$\tan(a + b) = \frac{\tan(a) + \tan(b)}{1 - \tan(a) \tan(b)}$

To determine the derivation of tan(2x), we make use of the following steps.

Rewrite 2x as the addition of x and x.

So, we have:

$\tan(2x) = \tan(x+x)$ --- step 1

Next, we apply the tangent sum identity.

Recall that:

$\tan(a + b) = \frac{\tan(a) + \tan(b)}{1 - \tan(a) \tan(b)}$

Substitute x for a and b.

So, we have:

$\tan(2x) = \frac{\tan(x) + \tan(x)}{1 - \tan(x) \tan(x)}$ ---- step 2

Simplify the denominator

$\tan(2x) = \frac{\tan(x) + \tan(x)}{1 - \tan^2(x)}$

Hence, the result of the derivation is $\tan(2x) = \frac{\tan(x) + \tan(x)}{1 - \tan^2(x)}$

Read more about trigonometric identity at:

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