Question

The values of $\theta \in \left[0,\frac{\mathrm{\pi }}{4}\right]$ satisfying tan 5θ = cot 2θ are ______________.

Answer 1

for $\theta \in \left[0,\frac{\mathrm{\pi }}{4}\right]$
tan 5θ = cot 2θ
i.e tan 5θ = $\frac{1}{\tan 2\theta }$
⇒ tan 5θ tan2θ = 1 ...(1)
Since
tan (x + y) = $\frac{\tan x+\tan y}{1-\tan x\tan y}$
i.e tan(5θ + 2θ ) = $\frac{\tan 5\theta +\tan 2\theta }{1-\tan 5\theta \tan 2\theta }$
i.e tan 7θ = $\frac{\tan 5\theta +\tan 2\theta }{0}$ (from 1)
i.e tan 7θ is not defined
i.e tan 7θ = $\tan \frac{\mathrm{\pi }}{2}$
⇒ 7θ = n$\mathrm{\pi }+\frac{\mathrm{\pi }}{2}$
i.e θ = $\frac{n\mathrm{\pi }}{7}+\frac{\mathrm{\pi }}{14}$
If $\theta \in \left[0,\frac{\pi }{4}\right],\theta =\frac{\mathrm{\pi }}{14},\frac{3\mathrm{\pi }}{14}$