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Trigonometric Ratios for Sum of Two Angles

The general s...

Question

The general solution of the equation $tan θ + tan 4 θ + tan 7 θ = tan θ tan 4 θ tan 7 θ$ is

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Solution

We know that, $tan (θ + 4 θ + 7 θ)$

$= \frac{t a n (θ + 4 θ) + t a n 7 θ}{1 - t a n (θ + 4 θ) t a n 7 θ}$

$= \frac{\frac{t a n θ + t a n 4 θ}{1 - t a n θ t a n 4 θ} + t a n 7 θ}{1 - t a n θ + t a n 4 θ 1 - t a n θ + t a n 4 θ + t a n 7 θ}$

$= \frac{t a n θ + t a n 4 θ + t a n 7 θ - t a n θ t a n 4 θ t a n θ}{1 - t a n θ t a n 4 θ - t a n θ + t a n 4 θ + t a n 7 θ - t a n θ t a n 4 θ t a n 7 θ}$

$t a n 12 θ = \frac{t a n θ + t a n 4 θ + t a n 7 θ - t a n θ t a n 4 θ t a n 7 θ}{1 - t a n θ + t a n 4 θ + t a n 7 θ - t a n θ t a n 4 θ - t a n θ t a n 4 θ t a n 7 θ}$

We know that $t a n θ + t a n 4 θ + t a n 7 θ = t a n θ 4 θ t a n 7 θ$

$\Rightarrow t a n 12 θ$ = $\frac{t a n θ t a n 4 θ t a n 7 θ - t a n θ t a n 4 θ t a n 7 θ}{1 - t a n θ + t a n 4 θ + t a n 7 θ - t a n θ t a n 4 θ - t a n θ - t a n 4 θ - t a n 7 θ}$

$\Rightarrow tan 12 θ = 0$

$\Rightarrow tan 12 θ = 0$

$\Rightarrow 12 θ = n π$

$\Rightarrow θ = n π / 12$

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Trigonometric Ratios for Sum of Two Angles

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