The number of values of θ∈$$(0$$,π] , such that $$($$$$cos$$⁡θ$$+isin$$⁡θ$$)($$$$cos$$⁡$$3$$θ$$+isin$$⁡$$3$$θ$$)($$$$cos$$⁡$$5$$θ$$+isin$$⁡$$5$$θ$$)($$$$cos$$⁡$$7$$θ$$+isin$$⁡$$7$$θ$$)($$$$cos$$⁡$$9$$θ$$+isin$$⁡$$9$$θ$$)=$$−$$1is$$

Question

The number of values of θ∈$$(0$$,π] , such that $$($$$$cos$$⁡θ$$+isin$$⁡θ$$)($$$$cos$$⁡$$3$$θ$$+isin$$⁡$$3$$θ$$)($$$$cos$$⁡$$5$$θ$$+isin$$⁡$$5$$θ$$)($$$$cos$$⁡$$7$$θ$$+isin$$⁡$$7$$θ$$)($$$$cos$$⁡$$9$$θ$$+isin$$⁡$$9$$θ$$)=$$−$$1is$$

Infinity Learn · Accepted Answer

Using $$($$$$cos$$⁡α$$+isin$$⁡α$$)($$$$cos$$⁡β$$+isin$$⁡β$$)=$$$$cos$$⁡$$($$α$$+$$β$$)+isin$$⁡$$($$α$$+$$β$$)We$$ getcos⁡(25$$θ$$)=$$−$$1$$, $$sin$$⁡$$(25$$θ$$)=0$$⇒ $$25$$θ$$=(2k+1)π,k∈I.Now, $$0$$<(2k+1)π$$25$$≤π⇒ $$1$$≤(2k+1)≤$$25$$⇒ $$k=0$$,$$1$$,$$2$$,…,$$12$$

The number of values of $θ \in (0, π]$ , such that $(\cos θ + i \sin θ) (\cos 3 θ + i \sin 3 θ) (\cos 5 θ + i \sin 5 θ) (\cos 7 θ + i \sin 7 θ) (\cos 9 θ + i \sin 9 θ) = - 1$
is

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The number of values of θ∈(0,π] , such that (cos⁡θ+isin⁡θ)(cos⁡3θ+isin⁡3θ)(cos⁡5θ+isin⁡5θ)(cos⁡7θ+isin⁡7θ)(cos⁡9θ+isin⁡9θ)=−1is

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The number of values of $θ \in (0, π]$ , such that $(\cos θ + i \sin θ) (\cos 3 θ + i \sin 3 θ) (\cos 5 θ + i \sin 5 θ) (\cos 7 θ + i \sin 7 θ) (\cos 9 θ + i \sin 9 θ) = - 1$
is