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  4. If sinθ = 3sin(θ + 2α), then the value of tan(θ + α) + 2tanα is

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Q. If $sin\theta = 3sin(\theta + 2\alpha)$, then the value of
$tan(\theta + \alpha) + 2tan\alpha$ is

Trigonometric Functions

Solution:

$sin\theta = 3sin\left(\theta + 2\alpha\right)$
$\Rightarrow sin\left(\theta + \alpha - \alpha\right) = 3sin \left(\theta + \alpha + \alpha\right)$
$\Rightarrow sin\left(\theta+\alpha\right)\,cos\alpha-cos\left(\theta+\alpha\right)sin\alpha$
$= 3sin\left(\theta + \alpha\right)cos\alpha + 3cos\left(\theta + \alpha\right)\,sin\alpha$
$\Rightarrow -2sin\left(\theta+\alpha\right)\,cos\alpha=4\,cos\left(\theta+\alpha\right)sin\alpha$
$\Rightarrow \frac{-sin\left(\theta+\alpha\right)}{cos\left(\theta+\alpha\right)}=\frac{2\,sin\,\alpha}{cos\,\alpha}$
$\Rightarrow tan\left(\theta + \alpha\right) + 2tan\alpha = 0$