The number of values of \theta in the interval \left(\right. \frac{- \pi}{2} , \textrm{ }\textrm{ } \frac{\pi}{2} \left.\right) such that \theta \neq \frac{nπ}{5} for n = 0, ± 1, ± 2 and tan \theta = cot 5\theta as well as sin 2\theta = cos 4\theta, is

Engineering

Mathematics

Two Equations in one Unknown

Question

The number of values of $θ$ in the interval $(\frac{- π}{2}, \frac{π}{2})$ such that $θ \neq \frac{nπ}{5}$ for n = 0, ± 1, ± 2 and tan $θ$ = cot 5 $θ$ as well as sin 2 $θ$ = cos 4 $θ$ , is

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$tanθ = \cot 5 θ \Rightarrow tanθ = \tan (\frac{π}{2} - 5 θ)$

$∴ θ = nπ + \frac{π}{2} - 5 θ, n \in I \Rightarrow θ = \frac{nπ}{6} + \frac{π}{12}, n \in I$

$∴ θ = \pm \frac{π}{12}, \pm \frac{π}{4}, \pm \frac{5 π}{12}$

Now $\sin 2 θ = \cos 4 θ \Rightarrow \cos (\frac{π}{2} - 2 θ) = \cos 4 θ$

$∴ \frac{π}{2} - 2 θ = 2 mπ \pm 4 θ \Rightarrow \frac{π}{2} - 2 mπ = 6 θor - 2 θ$

$∴ θ = \frac{π}{12} - \frac{mπ}{3} or θ = mπ - \frac{π}{12}; m \in I$

$∴ θ = \frac{π}{12}, - \frac{π}{4}, \frac{5 π}{12}$

$∴$ Number of values of $θ in (- \frac{π}{2}, - \frac{π}{2}) are 3 i . e . \frac{π}{12}, - \frac{π}{4}, \frac{5 π}{12} \Rightarrow 3$

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