Question

Prove tanx+tan4x+tan7x=tanx.tan4x.tan7x

Answer 1

Answer on Question #58052 – Math – Trigonometry

Question

Prove that $\tan x + \tan 4x + \tan 7x = \tan x \tan 4x \tan 7x$

Solution

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

Now consider $a + b + c = n\pi$, then $\tan(a + b + c) = 0$.

So,

$$\tan (a) + \tan (b) + \tan (c) - \tan (a) \tan (b) \tan (c) = 0$$ $$\tan (a) + \tan (b) + \tan (c) = \tan (a) \tan (b) \tan (c)$$

Therefore,

$$x + 4x + 7x = n\pi$$ $$12x = n\pi$$ $$x = \frac{n\pi}{12}.$$

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