By principle of mathematical induction $$cos$$⁡θ$$cos$$⁡$$2$$θ$$cos$$⁡$$4$$θ…$$cos$$⁡$$2n$$−$$1$$θ,∀n∈$$N=$$

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By principle of mathematical induction $$cos$$⁡θ$$cos$$⁡$$2$$θ$$cos$$⁡$$4$$θ…$$cos$$⁡$$2n$$−$$1$$θ,∀n∈$$N=$$

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Let $$P(n)$$:$$cos$$⁡θ$$cos$$⁡$$2$$θ$$cos$$⁡$$4$$θ…$$cos$$⁡$$2n$$−$$1$$θ$$=$$$$sin$$⁡$$2n$$θ$$2nsin$$⁡θ… (i) Step l :For $$n=1$$,          $$LHS=$$$$cos$$⁡θ and RHS $$=$$$$sin$$⁡$$2$$θ$$2sin$$⁡θ$$=$$$$cos$$⁡θ  $$P(1$$ $$)$$ is true. Step ll :Let $$P(n)$$ is true, $$thenP(k)$$:$$cos$$⁡θ$$cos$$⁡$$2$$θ$$cos$$⁡$$4$$θ…$$cos$$⁡$$2k$$−$$1$$θ$$=$$$$sin$$⁡$$2k$$θ$$2ksin$$⁡θStep III: For $$n=k+1P(k+1)$$:$$cos$$⁡θ$$cos$$⁡$$2$$θ…$$cos$$⁡$$2k$$θ$$=$$$$sin$$⁡$$2k+1$$θ$$2k+1sin$$⁡θ LHS $$=$$$$cos$$⁡θ$$cos$$⁡$$2$$θ…$$cos$$⁡$$2(k$$−$$1)$$θ$$cos$$⁡$$2k$$θ$$=$$$$sin$$⁡$$2k$$θ$$2ksin$$⁡θ⋅$$cos$$⁡$$2k$$θ$$=2sin$$⁡$$2k$$θ⋅$$cos$$⁡$$2k$$θ$$2k+1sin$$⁡θ$$=$$$$sin$$⁡$$2k+1$$θ$$2k+1sin$$⁡θ$$=RHS$$   For n $$=k$$ $$+$$ $$1$$, $$P(n)$$ is true. Hence, by principle for mathematical induction for all n ∈ N,$$P(n)$$ is true.

By principle of mathematical induction cos⁡θcos⁡2θcos⁡4θ…cos⁡2n−1θ,∀n∈N=

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