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  4. If θ ∈(π / 4, π / 2) and displaystyle∑n=1∞ (1/ tan n θ)= sin θ+ cos θ then the value of tan θ is

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Q. If $\theta \in(\pi / 4, \pi / 2)$ and $\displaystyle\sum_{n=1}^{\infty} \frac{1}{\tan ^n \theta}=\sin \theta+\cos \theta$ then the value of $\tan \theta$ is

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Solution:

$\tan \theta>1 \Rightarrow 0<\frac{1}{\tan \theta}<1$
$\text { now, } \frac{1}{\tan \theta}+\frac{1}{\tan ^2 \theta}+\frac{1}{\tan ^3 \theta}+\ldots \ldots \infty=\sin \theta+\cos \theta$
$\frac{\frac{1}{\tan \theta}}{1-\frac{1}{\tan \theta}}=\sin \theta+\cos \theta \Rightarrow \frac{1}{\tan \theta-1}=\sin \theta+\cos \theta$
$ \Rightarrow \frac{\cos \theta}{\sin \theta-\cos \theta}=\sin \theta+\cos \theta $
$\cos \theta=\sin ^2 \theta-\cos ^2 \theta=1-2 \cos ^2 \theta $
$2 \cos ^2 \theta+\cos \theta-1=0 \Rightarrow (2 \cos \theta-1)(\cos \theta+1)=0 $
$\cos \theta=\frac{1}{2} \text { or } \cos \theta=-1 \text { (rejected) } $
$\theta=\frac{\pi}{3} \Rightarrow \tan \theta=\sqrt{3}$