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Question Number 67664 by Rio Michael last updated on 29/Aug/19

show that ∃ n ∈ N^(+ ) : sin^n x + cos^n x = 1 and cosh^n x − sinh^n x = 1. Hint: use Induction method.

$${show}\:{that}\:\:\exists\:{n}\:\in\:{N}^{+\:} \::\:\:{sin}^{{n}} {x}\:+\:{cos}^{{n}} {x}\:=\:\mathrm{1}\:{and}\:\:{cosh}^{{n}} {x}\:−\:{sinh}^{{n}} {x}\:=\:\mathrm{1}. \\ $$$$ \\ $$$${Hint}:\:{use}\:{Induction}\:{method}. \\ $$$$ \\ $$

Commented by Rasheed.Sindhi last updated on 30/Aug/19

2∈N⇒sin^2 x+cos^2 x=1

$$\mathrm{2}\in\mathbb{N}\Rightarrow\mathrm{sin}^{\mathrm{2}} \:{x}+\mathrm{cos}\:^{\mathrm{2}} {x}=\mathrm{1} \\ $$

Commented by Rio Michael last updated on 30/Aug/19

thanks but sir do we have any better way to prove?

$${thanks}\:{but}\:{sir}\:{do}\:{we}\:{have}\:{any}\:{better}\:{way}\:{to} \\ $$$${prove}? \\ $$

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