If $$cot3\theta \cdot cot6\theta = 1$$, then the value of $$tan15\theta$$ will be​

Correct option is C

Given:

cot⁡(3θ) cot⁡(6θ) =1

Concept Used:

We know the identity:

For,  $$A+B= \frac{\pi}{2}$$  

​$$\cot A \cdot \cot B = 1$$​​

Solution:

3θ + 6θ = $$\frac{\pi}{2}$$ 

9θ = $$\frac{\pi}{2}$$ 

​θ = $$\frac{\pi}{18}$$ 

Then, value of tan⁡ (15θ) 

θ = $$\frac{\pi}{18}$$ = 15​θ = 15 $$\times\frac{\pi}{18}$$ = $$\frac{5\pi}{6}$$ 

Then, 

​$$\tan\left(\frac{5\pi}{6}\right) = -\tan\left(\pi - \frac{\pi}{6}\right) = -\tan\left(\frac{\pi}{6}\right) = -\frac{1}{\sqrt{3}}$$ 

​So, the value of $$\tan(15\theta)$$​ is: $$-\frac{1}{\sqrt{3}}$$ 

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